Due to a number of reasons, I would like to keep my freewheel at all costs (It's a Sachs freewheel, the bike has all sorts of more 'vintage' components, and they're hard to come by). From reading all the related questions, it seemed the solution would be a new chain, and the teeth certainly don't look worn enough (to me at least) to warrant the freewheel needing replacement.
However, with both a brand new 7/8 speed chain, and the older slightly worn chain (0.140 in elongation over 12 links), the chain rides up as shown on the teeth when torque is applied. It doesn't happen in the highest gears, or when it's upside down and I'm adjusting derailleurs. The chain is also not touching any adjacent gears due to derailleur misadjustment.
As stated, I will try ANY reasonable options before replacing the freewheel. I've certainly ridden more worn out cogs without this problem. Should I just try a different new chain? Or is this more seriously worn than I judge it to be?
Additional Pictures added on request: