Maybe the area has more crap on the ground that causes flats than other areas, because it sounds like a lot of flats for so short a time period. When you flat, take a look at the tube to see where the flat is; is it facing the road or the rim? Is it on the side? A small hole, a tear, a double hole (snakebite). Where and how the hole looks can tell a lot about what the cause is, then you can respond with whatever is needed to deal with it.

If it faces the rim, your rimstrip should be replaced, it's either worn or may have moved off of the spoke holes, exposing the metal edge to the tube. Facing the side; tube may be pinched between the tire and rim-like snakebite. Facing the road; check the tire for the cause, if it's still in the tire it will strike again!

Yes a worn tire will flat more, as there is less material on the tire to help prevent something from pushing through the casing. Hopefully your changing the tires will lessen those pesky flats, but flats will happen again. Just be familiar with the patch kit and keep it handy. Take a pump; I'm no fan of CO2, once you use it it's gone and if you then flat again you are SOL.

Part of my fear of riding alone is handling technicals, so I made sure I know how to adjust brakes/shifters, can change tubes/patch flats, carry a multitool that can handle all the 'nuts 'n bolts' on the bike, and take a map if in unfamiliar territory. I often ride alone (as I did today on an MTB ride in the Santa Monica mtns.) To improve general fitness, I've been commuting to work and that has improved my fitness, calmed my fear of riding on a busy street, and even got me acclimated to riding in the dark with lights (and to look ahead, not down at the street in front of my front tire, which I can't see in the dark anyway.)