I have a few thoughts on this. My partner had similar issues after a long hiatus from cycling. She's roughly 5'3".
- I'm not sure a mountain bike was the best choice for a beginner. Something with a step-through frame would probably be better as it would give you a more direct line of escape if necessary. That said, I think the bike is probably a rational size for you if it's the woman's version but if it's a man's version, the reach (basically the distance to the handlebars) may be too long for you. As others have noted, it's important to get the seat and handlebars adjusted properly and, depending on how mechanically inclined you are, there are modifications you can make to the bike to provide further adjustment. I'm not sure swapping stems and such, without some experienced advice, is a good idea, because putting a shorter stem on a bike than the bike was designed for can affect the steering and handling of the bike making it feel even less stable. swapping the handlebars for something with more back sweep is a good suggestion as well as seeking out your local bike co-op.
- also be sure your tires are properly inflated. underinflated tires, especially those big 2" mountain bike tires, and when moving slowly, can cause an unstable squishy feel on pavement. it can also cause a blowout.
- As for gears, you probably don't want to do too much shifting until you gain your balance so put them both in the middle for now. If it's too hard to pedal, downshift with your right hand. Find a gear that works for taking off but doesn't leave you spinning your pedals once you get going and just leave it there. When you gain more confidence, you can start learning how to shift. Basically, the idea of shifting gears is to adapt to varying terrain while trying to maintain a consistent pedaling cadence and level of effort. When you're in your cruising speed gear and you start going down hill, you shift to a higher gear so you can keep pedaling without spinning out. As you start going up hill, you shift to a lower gear to make it easier to climb the hill.