At least in the Royal Navy in the period I study (roughly the Napoleonic wars), drunkenness was in theory punishable by a court-martial -- it's an offense that's specifically mentioned in the RN's Articles of War, an online version of which is hosted here. Drunkenness actually finds itself inserted into the second of the Articles, the first one being a mandate to carry out Anglican service in their ships. You can read the relevant part here:

All flag officers, and all persons in or belonging to His Majesty's ships or vessels of war, being guilty of profane oaths, cursings, execrations, drunkenness, uncleanness, or other scandalous actions, in derogation of God's honour, and corruption of good manners, shall incur such punishment as a court martial shall think fit to impose, and as the nature and degree of their offence shall deserve.

What's important about that particular passage isn't so much that it makes drunkenness per se a court-martial offense, but that it applies to "flag officers, and all persons" -- that is, even commissioned officers could theoretically face a court-martial for drunken behavior.

This was in a time when alcohol was served fairly freely on British ships; officers and warrant officers would drink wine or other spirits with meals, and there was no prohibition against keeping private stocks aboard. Ordinary sailors were in theory limited to their ration (a gallon of beer a day in home waters and a half-pint of rum or other spirits on foreign service) there was definitely an illicit trade in alcohol belowdecks. Regardless, simply being drunk was not usually enough cause to punish a man -- being drunk to the point of insensibility and failing to perform duties (e.g. being unable to go on watch) might be, though. Punishment could vary, but anything from watering the grog ration, to stopping grog, up to floggings for repeat offenses could be a possibility for sailors. Captains had a wide range for punishment; the Articles of War close with what was called "the captain's grab bag:"

All other crimes not capital committed by any person or persons in the fleet, which are not mentioned in this act, or for which no punishment is hereby directed to be inflicted, shall be punished by the laws and customs in such cases used at sea.

I can shed a bit of light on your other question, on the background of the shanty, or perhaps only obscure the light it once had.

The melody is the same, or nearly the same, as an Irish Gaelic piece known as *Óro ’Sé do bheatha 'bhaile * (with apologies if the accents are screwy; my limited Gaelic is the other kind) that has old Jacobite lyrics that seem to go back to the '45.

A Shéarlais Óig, a mhic Rí Shéamais

'Sé mo mhór-chreach do thriall as Éirinn

Gan tuinnte bróig' ort, stoca nó leinidh

Ach do chascairt leis na Gallaibh

I can't really attempt to reliably translate most of that with non-existent Gaeilge (/u/depanneur, can you?) and don't trust the online translations I can find, but the first line is definitely referring to "Young Charles, King James' son". However, I checked the date of this tune against the date for Drunken Sailor, and it seems that the attested date for the Irish one is a couple decades later than for Drunken Sailor. So while the lyrics might possibly go back to 1745, we actually don't know that, or know whether they were sung to the same melody then.

And to quickly complete the story, Padraig Pearse rewrote Óro ’Sé do bheatha 'bhaile in the years before the Easter Rising, removing references to the Jacobites, and this is the version you almost exclusively hear (Link has the lyrics up, too, in case your Gaeilge is better than mine).