Never had a problem with Hornady 38/357 sizer on a Hornady press in 15 years. Like you say, really weird to happen all of a sudden. Hornady press? Progressive press where something moving got worn or a new piece put on? Hate it when stuff just happens: mostly because when I figure it out it's some little thing I changed and forgot that I did it
Lots of good things to try posted above. Asking Hornady would be worth your time once you give up on DIY.
Spitballing here: Do you just dry-tumble brass? I've read that at least with some steel dies (NOT carbide), wet-tumbling brass really clean could cause them to stick a little.
Nebarsa-resident commercial break: My favorite cleaner/lube, especially where a dry lube is better, is Hornady OneShot Gun Cleaner and Lube. I use it on firearms on initial cleaning and occasionally after that, but always on my reloading dies and presses. Probably won't help as much on the carbide part of the 38/357 die, but good for press moving parts. Won't contaminate anything gun-related and is resistant to getting gummed up.