Was he providing child support previously? You say, "I'd like to know where he's living so I can have an attorney send a letter asking if he's planning on being involved or would just terminate his rights." Is the crux of the matter really that, or is it getting child support or some other thing? Obviously he's made some very impulsive decisions but has a long history of a relationship w/ your DD. I don't know that I'd want to choose this time to ask whether he just wants to terminate his rights. His little honeymoon break from reality may not last forver; in fact, the more impulsive & quick this marriage was, the quicker it may bellyflop. If it's just a matter of "are you in or are you out?" I'd rather give him time to come to his senses & not ask the question yet & force the choice. I know that 5 months seems forever to your DD and that you'd just like some closure one way or the other, but in the long run she's probably better off if he doesn't terminate rights...and if you give him some space to come to his senses on his own (which is usually more effective than trying to convince someone that they've gone off half cocked). If you are hoping that the letter from an attorney would get him to realize that he needs to start seeing his kids again...well, again, I'd give it some more time before I forced his hand. I would focus on finding another source of free or reduced price therapy for your DD. Parental rejection can certainly be hard on a child's self esteem. I hope that the man you're in a relationship with now, has a good relationship with her.
Now, if you had been receiving child support & desperately need it, that puts things in a different light.
Don't assume that he won't use his social; don't assume anything, since he's already proven that he's capable of doing something completely stupid and hasty. He got married & moved very quickly; haste makes for mistakes in covering your trail. Do you have any idea that he might have moved out of state--and might need to get a new driver's license? Hmm, if he got married, there's a marriage license somewhere, which would give you the wife's DOB, and with a first name & DOB it's easy enough to do a search on her. Find the marriage license. Find the marriage announcement in the paper, if there was one, and go from there; for example, you can mine it for the names of her relatives. Where did he live before? Did he leave the landlord or previous employer a fwding address? Was he silly enough to file a change of address form--or did she? If you assume anything, assume that he made some kind of stupid mistake and that you WILL find it.
Short of a private detective, there are people on the internet who have subscriptions to various search databases and who'll run a search on his soc, driver's license number, etc. for a much more reasonable price. I haven't needed one for about 10 years but try going into yahoo groups and finding a group for people who are searching for their ex. Lurk (just read) for a while and try to get a sense for what people are like, and you can also pick up some search tips that way. Be careful to use someone that has been helpful to other people in the group, not just anyone who says they'll do a cheap database search.
If you are low income you can see if Legal Aid for your county has any suggestions.
Good luck! Again, if all you want to do is give him an ultimatum about terminating his rights in the hopes that he'll see his kids or at least give you some closure, I think I'd wait it out for quite a while longer. It would be handy to have his location, though, while the trail is still warm.