Sorry, I don't usually post on the weekends and I was busy yesterday. But I experimented a little at practice last night and I think I know what it is.
My guitar is a Vantage Gibson 335 rip-off, not a Strat. It has no tremolo. But I think that when I bend the neck back or maybe even bend the string, the string is catching somewhere and thus staying sharp. So I guess I need to do the ol' pencil lead trick. I'll let you know if that fixes it.