Generally, yes you can do so, as long as you pay the full price for the new BlackBerry. (So it's probably best to go to a place like eBay or something, since you probably won't be able to get a discounted BlackBerry unless you've been with Cingular for long enough to qualify for a HUP)
Theoretically, you can probably even resell your old phone, but you are still stuck under the same contract. (Although ethically, I don't know about that, though this is probably OK if you continue honour the existing contract, and are currently under no obligation to ever return the phone.)
Changing plans is probably allowed as long as it's a higher-priced plan, but you should check with your carrier first. Basically, you'll probably have to keep your existing plan, but add $45-per-month for the addition of unlimited BlackBerry data service. You should call Cingular first to ask if you can enable BlackBerry features on your SIM card.
A great place to ask about cellphone carrier questions is HowardForums at
http://www.HowardForums.com. Check out the Cingular forum there. Also, there's the
AT&T Wireless Support Forums although I don't know if that now covers Cingular too.