Thanks for all y'all's help! I greatly appreciate it
I didn't notice any black smoke, but that's very possible since I was testing it at night and wasn't looking out for it. I did smell and thought I saw a little smoke for a second, though.
I'm pretty sure the carbs are good! @
ducatiman's work was fantastic; the immediacy and reoccurrence of the issue make me think that it wasn't the carbs (maybe a setting issue?) or the gas tank (I double-checked that, too, and it looked good; maybe one or two specks of flash rust that were caught in the in-line filters) and is probably something else wrong with the bike. The petcock is very likely.
Will do on examining the battery! I'll charge it before I try starting it up again.
I'm probably going to have to do a valve adjustment. The bike's at about 10,100 miles. Considering how the last owners maintained the bike (the tires are 10 years old, which I'll have to change once I get it running), I highly doubt it's been done.
Could the air filter be an issue? That's the only thing I can think of to check other than the valves.