Taking Livy as an example, are historians using Livy as a jumping off point because there are no other primary sources, or is it more a case of looking at archaeological findings and trying to create a narrative independent of Livy?
This is a very difficult question to answer, but I will do my best to keep it succinct. There are a few ways in which modern historians test the veracity of their ancient sources:
One will need to determine a few things to try and reach a conclusion:
A. Any biases of the authors B. Chronological distance from events C. The ancient historians' own sources
These are just three of many helpful paths to determine which author is speaking the truth.
What to do if you want to verify Livy, but there are not corresponding sources? Now, this gets a bit tricky.
As you said, archaeology can be helpful for us here. If Livy tells us that town X was destroyed by the Romans on Y date, we can attempt to locate the ancient site and search for ruins. This is a useful way of corroborating ancient sources.
Another very important method is using inscriptions. This is a whole sub-discipline unto itself called epigraphy. We find hundreds if not thousands of new inscriptions from the Romans world every year. Now, the vast majority of them are banal, but some are quite fascinating. Some may even come to corroborate or invalidate an ancient historians' claim.
You may not like this answer, but sometimes a modern historian must use his gut. Unfortunately, a paucity of sources means we have to rely on our scholarly intuition and good judgment. Some scholars tend to be rather pessimistic about the ancient historians, while others tend towards the optimistic. I have yet to really formulate my own thoughts on he topic, but I will leave you with a piece of advice that one of the greatest Roman historians of our generation once told me: "We, historians of the ancient world, have so few sources, but many of us spend most of our time trying to explain them away."
P.S. this is not meant to be an exhaustive account of every way modern historians try to prove the veracity of their sources, but rather a concise recounting of some of the more prominent methods.
Many of us approach these sources as literature rather than what you'd strictly call 'history'; something like historical fiction, or perhaps more accurately like editorial opinion pieces about historical events.
Just as modern political editorials often talk about history (and often play pretty freely with the actual facts, because they're really mostly interested in their pieces' political implications), ancient authors were almost always using the past to talk about contemporary events and concerns. As a result, it's very difficult to trust their stories to be objective, accurate, or faithful descriptions of actual events. They're onspired by actual events, but unless you have other information available about those events, it's often impossible to tell how much the author's literary / political objectives have shaped the underlying 'facts' of the matter.
That is, it's not just a matter of reading through the bias to see the facts underneath; the facts themselves are, on their own, so tied up in the authors' projects and goals that they all have to be taken with a large grain of salt.
That's why the best historians do things like /u/gloriasinefine describes to try to verify accounts externally. But even then, you have to be careful with textual sources - just because an archaeological discovery proves, fkr example, that a battle happened doesn't mean the author who describes the battle understood why it happened.
Ie, we knoe that the Punic Wars happened. But we should still question Livy's description of Hannibal, to see how his version of events fits into his larger goal of glorifying Augustus' Rome.
The best historical studies takes multiple types of sources (narrative texts,mrecords,minscriptions, coins, archaeology, art, geography, demographic modelling, climate data, etc etc), reads each kind of evidence independently and critically, and then weaves these different pieces together to capture the complexity of past events.
I personally use texts to add color to my work - they provide a perspective on events from at least one contemporary person,s perspective. Just like good journalists quote eyewitnesses to make their work more exciting, but rely on careful contextual research when maki their actual analysis of events.
This thread may also be helpful: https://www.reddit.com/r/AskHistorians/comments/3uvva1/monday_methodsfinding_and_understanding_sources/