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Sometimes, "auto-shifting" can be caused by frame flex, which also occurs predominantly under hard efforts. But that's not a problem that would have just manifested itself, unless your frame has cracked or been damaged. It's worth a check just to be safe.

You can check the chain by measuring. On a new chain, you should be able to measure exactly 12" from center to center of 12 inner/outer link pairs. On a worn chain, if it measures 12 1/16", it's time to change it. If it measures 12 1/8", you're overdue and wearing out your drive train.

A lot of people complain about gear wear on the freewheel. I notice it more on the chainrings (maybe it's a roadie thing). Either way, if that's the problem, the new chain will make it worse (since they've worn adapting to the "stretched" chain).