How do I know if my eyeglass prescription is wrong?

Well first of all, I just got vision insurance and I'd been wearing the same glasses for about 5 years before my insurance. My previous prescription was obviously too weak. I couldn't read signs, or my teleprompter at work well even with them on. So I got my eye exam, and a new prescription.

Well I just got the glasses today, and I'm seeing really well. From just a vision standpoint they seem to be perfect. But I'm having a little trouble. It's hard to explain, but my the frames are sort of small, and my peripheral vision is kind of messing me up. You know how if you have a glass of water, and stick a glass rod in it, there's a bit of a disconnect and the out of water and in water parts of the rod don't quite match up? That's what I'm seeing right now. Walking feels a little weird and it makes me a little dizzy, because unless I'm staring straight at the ground, the vision in front of me is clear, and my peripheral is blurry, but the size of the two don't match up, and I feel like I'm about two feet tall trying to walk in a body that isn't.

Is this normal for the adjustment period of glasses? Will I get used to it and learn to (for the most part) ignore my peripheral vision? I do know that I have very sharp and aware peripheral vision. Or is my prescription perhaps over correcting my eye site a bit too much? It's really hard for me to tell. Like I said, I feel like I'm seeing really well; I'm seeing details that I haven't in years. But I'm also used to wearing such a weak prescription....maybe I just don't know any better.

Other symptoms? None really. I haven't noticed any sort of headache or general dizziness (other than what I've described). My eyes do feel a little strained, but like I said, I've been in the wrong prescription for years, and my eyes certainly aren't used to seeing at 20/20.