Ok, first things first:
If you bought it overseas, it's most likely GSM. If it's unlocked, it's almost definitely GSM. If it's quad-band, it's certainly GSM. Unlocked GSM phones can be used with Cingular, T-Mobile, Dobson, and any other GSM carrier. Sprint and Verizon (and Alltel I think) are CDMA carriers, and must use CDMA phones. They aren't interchangeable: you can't use a GSM phone on a CDMA service.
GSM phones are never analog. You insert your SIM card from your previous phone into the new phone, and it's basically like plugging a new phone into your "phone jack." It will still connect to the same carrier, whatever their signal strength may be in your area.
An "unlocked" phone basically means you can use it with any SIM card you want. The SIM card is what determines what carrier you connect to. And GSM phones must be quadband from the factory, they can't be field-modified to become quadband.
Since you state you use Verizon, you can't use it. Verizon does not use SIM cards, and Verizon has a habit of only activating Verizon phones on their network. Sprint is the same way...it's common to CDMA carriers in the USA. CDMA phones in the USA are all dual-band digital, and some also offer analog. SO even if you had gotten a CDMA phone off of Ebay, likely you couldn't use it unless it was already a Verizon phone. GSM phones are locked on the phone side, CDMA phones are locked on the network side.
Basically, you're stuck with your phone. Verizon does not use SIM cards, because it's a different technology, and then they wouldn't be able to charge you every time you call them up to change phones.
