Hillary held a very strong Republican view towards gays for most her her career, i.e. she was anti-gay rights, despite the Constitution's Bill Of Rights. It was her husband that signed the very Republican sounding Defense Of Marriage - DOMA - act (which, of course, he regrets). They were successful in protecting marriage from gay people so they clearly became very close to Republicans. Well, at least some of them. (I sometimes wonder if 'moving to the center' is something the rich people who own media came up with to gain consensus to push their own agenda through and then started acting like is was a real rule... like they seem to do with "news" even today)
As is the pattern of the Clinton's, they make bad decisions and then they change positions and act like they made the right decision all along. Unless they are left with no choice but to apologize because what they did is just so bad that they have no choice. They have made some good decisions on the small scale (& the Democrat base is immune to religious economics... so there is a limit to what they can pull off there... not that they aren't able to, just takes more work). Since Democrats follow polls and Republicans create them, some of this may simply be coincidence.
Here is Hillary Clinton Against & For Gay Rights (Interestingly enough she's calling it a 'Constitutional right' now. Took her decades to figure that out. Definitely not a plus for her decision making skills):
From The Atlantic: