Most of the states who decriminalized sodomy during the 60s and early 70s did so as part of a broader re-writing of their criminal codes. Many states had incredibly complex and outdated criminal laws by then, with old laws against a variety of offenses which were no longer prosecuted or which courts had rendered moot still on the books. During the 1950s, an organization called the American Law Institute (ALI) drafted a "Model Penal Code" which it hoped states would use to create a new, more streamlined and more up-to-date criminal code.

The model code excluded any sexual offenses that were committed in private, between consenting adults, primarily because the legal experts who drafted it were of the opinion that such acts were in the realm of religious, rather than legal morality, and didn't harm the "secular interests" of the community. So the model code didn't just leave out sodomy, but also a variety of older prohibitions against oral sex, adultery, etc.

Illinois was the first state to use the model code as a guide when it updated its criminal code in 1961, adopting whole sections of it without much scrutiny. So it was the first state to decriminalize sodomy. Several other states followed later in the 60s and 70s.

More generally, though, it sounds like you're under the impression that there wouldn't have been an influential "gay rights" movement during this period, which is not correct. The gay rights movement as we know it began to flourish quite dramatically during the late 60s and early 70s - and gay rights activists did lobby hard for the decriminalization of sodomy throughout this period.