Q: Can my car drive 350 miles with this issue?

Have the battery load tested. If you go to a big battery store, they will test the battery for free. If your battery is 5 years old or older, there’s a good chance you need a new battery. Indeed, if you know that the battery is 5 years old or older, I wouldn’t bother with a load test; I would just replace the battery outright. Batteries are a VERY important component in a car’s electrical system. If the battery passes a load test be sure it is NOT a marginal pass in which case you absolutely should install a new battery given the particular symptoms that you are describing and to reduce diagnostic headaches. Once you are confident you have a good (or new) battery, test charging system output which should be around 14.5 volts. If the serpentine belt is original, don’t bother inspecting it, just replace the serpentine belt outright. If, however, at the moment you are just trying to travel 350 miles to complete a trip, visually examine the belt. If the belt is tight (appropriately tensioned) and you do not see any obvious signs of fraying, separation, and so forth you are probably OK for that distance but do get the serpentine belt replaced "soon" if it is original because you are due at 100,000 miles. If you have further questions or concerns, do not hesitate to re-contact YourMechanic as we are always here to help you.