There's definitely truth in what Jackson is claiming about the Second Bank. More context helps paint a clearer picture, but the Second Bank did try to sway the election of 1832, and was definitely engaging in some level of misconduct.

Alittle on calling Jackson a populist/propagandist: when looking back on banking in the early 19th century, people run a risk of making the common mistake of applying post facto knowledge and understandings to historic actors. While today concepts like a central bank, or of fiat currencies, seem natural, in the early 19th century these ideas were treated skeptically. Washington said that "Paper money has had the effect ... that it will ever have: to ruin commerce, oppress the honest, and open the door to every species of fraud and injustice." John Adams, a federalist, stated "Every dollar of a [paper] bank bill that is issued beyond the quantity of gold and silver in the vaults represents nothing and is therefore a cheat upon somebody." Many similar opinions can be found in newspapers, and commonly in the letters and opinions of businessmen. We can look back on this and scoff, in an era with FDIC insuring our deposits and with our currencies stable.. but within the world they knew, they were skeptical for good reason.

So why the skepticism towards banking? Well, the federal government at the time only concerned itself with making one kind of money: coins, made from precious metals. There was no official paper currency of the federal government. Instead, banks would keep precious metal coins (or specie) in their vaults, and loan out paper notes redeemable for specie. Furthermore, banks could loan out more paper than they had specie to cover, which was known as fractional reserve banking. So long as there was never too much specie demanded at once, this was a way of expanding the amount of money in circulation. Which was necessary, because the United States had a big problem during this era: it simply could not retain muchspecie. The economy was rapidly expanding and new cash was sorely needed. But, for a variety of reasons, specie had a tendency to wander off, mostly to China.

This means you had many banks, who had lots of credit to issue, and not nearly enough specie to actually cover the notes outstanding. This massive expansion of credit, combined with a crash of food production in Europe, led to a massive speculative bubble in western and southern lands which popped in the Panic of 1819.

And this is where the second bank enters the story. The second bank is chartered in 1816, when the bubble was starting to really expand. The bank was chartered for many things, but for this story the important parts are that a) it helped expand credit at first, which helped inflate the bubble even further before its crash, and b) it was chartered to collect all debts for the government, which it should collect in specie, in order to curb the excessive lending of the banks deeply involved in this bubble. Of course, when the second bank started really demanding payments in specie, it led to a huge crash in the lending sustaining the bubble, and contributed to it popping.

Now, it would be wrong to blame the second bank entirely for the panic. However, it did play its role in expanding credit, then strangling credit right when the bubble was bursting. And the effects of the panic were fairly disastrous. This is prior to when rigorous econometric data was taken, but a few stats have been pieced together about the effects of the panic of 1819: it appears that almost 50% of workers in Philadelphia and Pittsburgh were unemployed. Nearly half of the land that was sold in Alabama in the past 10 years, a substantial amount, had to be repossessed by the government. As a result of the bankruptcies, many people had to move into territories and live off the land, homeless and destitute.

Simply, 1819 was very, very bad for western and southern farmers who had got caught up in the speculative frenzy. Larger landowners were able to weather the storm, and most of the northeast banks were able to weather the storm. Naturally, alot of resentment was bred amongst these people against banking, with the second bank being a central locus of this resentment. These people would later become the popular base of Jackson.

It should also be noted in passing that banking wasn't regulated in any of the ways modern banking is regulated today. Bankers, including at the second bank, would engage in practices today seen as obvious fraud, but were not illegal then. For instance, specie could trade at a premium higher than the paper the bank gave out, and when it did bank directors would trade themselves at a 1:1 ratio their paper for the bank's specie. These types of 'legal but obviously morally wrong' acts did not help the perception of the second bank.

So when Jackson is opposed to the second bank in the ways he was, it's important to recognize that he wasn't totally out of whack with the popular prejudices of his day. It's become unfortunately rather common to 'psychologize' Andrew Jackson and point to earlier financial trouble he had as the locus of his disdain for the second bank, but in many regards he had parity with popular biases about the bank. So while I'd say it's definitely possible to describe him as a populist, I wouldn't say these attitudes constitute 'propaganda', which implies a sort of top-down tricking of the population into believing something new. Those attitudes formed organically without Jackson's aid, alongside Jackson himself.

Okay, let's now discuss the specific claims you quoted. Skip ahead to 1828. Andrew Jackson wins the election in a landslide, mostly supported by the same people who suffered in 1819. Shortly after the election, Jackson hears isolated reports that local agents of the Second Bank had spent bank money supporting the campaign of John Quincy Adams, Jackson's 1828 opponent. Jackson is very concerned by news of this, and soon democrats from a couple regions are coming to Jackson with similar tales.

This is one of those unfortunate cases in history, as with the Aaron Burr conspiracy, where it is impossible to confirm the validity of the accusations. There is simply no hard evidence either way. Is it possible that bank members did do this? Sure. People have embezzled money for various purposes before, it's not hard to imagine someone using some to help their preferred candidate. Is it also possible that the democrats, and Jackson in particular, were prone to believe rumors without enough skepticism? Yes, and it definitely seems democrats of the era were prone to this type of rumormongering. As were most people, to some degree. The Republicans would later come to believe a conspiracy that Jackson was trying to empower New York banks, which had as little evidence as the democrat claims against the bank. Regardless of valdity, though, Jackson and his cabinet took the claims very seriously. Complaints were brought to the Second Bank and an investigation was requested.

This is where we can introduce the primary antagonist to Jackson in the bank war: chairman of the Second Bank of the United States, Nicholas Biddle. Nicholas Biddle was a competent bank administrator, and was, like Jackson, prone to a bit of pridefulness, and an unwillingness to back down from a fight. To Biddle, the suggestion of impropriety in his organization was not acceptable. Despite initial reticence, friends of Biddle encouraged him to do an investigation, if only for reasons of perception. This Biddle did, he traveled personally to the branches accused and ran a very brief investigation.. which concluded no improprieties had happened. Soon after Biddle left the chastizing remark that the whole claim was "paltry intrigue got up by the combination of small bankrupts & smaller Demagogues". Clearly an insult to Jackson!