"Do not try to "lose"your passport, since the State Department puts you on a list and if it turns up later on an illegal immigrant you are in deep poopy.... "
Hillbilly - sounds to me like there is a story to be told regarding this one!
Get a passport asap - as said before, if nothing else everytime you look at it, there are great memories associated with those little stamps~ Mine expired last year (2 days after arriving back from DR) and I am sad that now I must get a new one before we head out on vacation this year. That new one won't have all those memories associated with it.
The Department of State website should answer all your passport questions - check on the travel announcement pages for any specific things you may need for travel to DR, according to it, you are ok with birth certificate and picture id -
http://travel.state.gov/passport_services.html