Please show where he "bragged" about signing DOMA into law. He refused to even to sign it publicly. Most Presidents make a big press deal out of bills they sign that they are "proud" of. Heck, so do governors, Governor Pence providing the perfect example with this law. And there was plenty of support at that time for a Constitutional Amendment. Had Clinton not signed it, it is more than possible and even very likely that Congress would have passed a Constitutional Amendment. With DOMA in place, til support for same sex marriage could increase (and anyone with half a brain could tell even at that time which way that was headed), it was a way to stave off a Constitutional Amendment making it much harder to get repealed.
And DADT was a compromise at the time so that recruiters couldn't actually ask those trying to join up if they are gay/homosexual (and yes, that was asked, and completely legal to turn someone away for answering before). It also didn't allow people in the military, including a person's chain of command to ask if someone were gay/homosexual and have that legally held against them. It made such a question illegal for the military to ask of a servicemember so that gay service members could actually serve, without lying about it, so long as they didn't come out and declare themselves gay/homosexual in a way that was a matter of public record. It also allowed for Clinton to remove the restriction on being gay as a qualifier for serving in pretty much all federal government jobs, including making it against the law to use as a reason to deny a person a security clearance of any kind.