I recently bought a "Schwinn Women's Clear Creek 28"/700c Hybrid Bike", and it turns out I'm uncomfortable with the wheel size. I couldn't figure out why I'm so unstable on it. Even though I haven't ridden a bike in 10 years or so I should have been able to pick it up right away. Well, I found out that the bike I rode from the time I was 10 until I was in my early 20's had 24" wheels. It was also a mountain bike. This new bike is a hybrid bike and has 28" wheels. With my old bike I was comfortable and in control when my seat was higher than the handlebars and I was still able to touch the ground. With these 28" wheels, my seat is up as far as I can go with my feet still able to touch the ground and I don't have enough height over my handlebars to feel stable (and I'm 5'11"). Does anyone know if it's possible to change out these wheels for 24"? Or is that size only for kid's bikes or mountain bikes? What about 26"? Thanks!
Changing the wheels will not change how you sit on the bike or the "fit". This bike uses rim brakes and a smaller wheel will force you to modify them heavily if it all possible. You can't wing this one and doing it right is way more expensive than this or a proper fitting bike is worth.
Hybrid bikes are made to be more for a upright seating position. That is why you can't really get your seat higher than your handlebars. They also tend to put the handle bars closer to the seat making for a shorter reach. The wheel base may be shorter as well which makes them more twitchy.
The idea that you shouldn't be able to reach the ground seated is because while seated, your leg should almost fully extend on the down stroke. This saves your joints while being more comfortable and efficient.
I have had to commute on my wife's hybrid bike while my mountain bike was in the shop for a few days, and yes, it is more twitchy so it is something that you have to get used to or find a bike that fits you better. 700C wheels also has some advantages over 24 inch that you are already enjoying but don't know it yet, but would notice on a downsize.
With the rims brakes your bicycle use I would say there is no way you can change the wheel to a smaller diameter. But that is not a problem.
You say last time you rode a bike was more than 10 years ago, and it was a different type of bike. It makes sense that you find it hard on this new bike, because more than it being different, you haven't been doing it for a while. I ride bikes on a daily basis and if I change my bike to a friends it takes me some time to adapt to the new bike. This is normal, and is more true as different the bike is (for example, road bike to mountain bike).
There is a chance you have bought a bike which is bigger than it should be (bikes have sizes), but this should also not be a big problem. If you give me your height and your frame size I can help you though.
I would however say there is nothing wrong and you bought an excellent bike, you just need to train a little bit. Lower your saddle to a comfortable position, and spend an afternoon training in a big, traffic free area. You will see you will re-learn your skills and be comfortable in no time.
Happy riding and remember, don't forget you helmet :)