Got a new chain, for ten speed rear cassette. Slightly shorter than old chain, but can still reach large sprocket, large cassette ring, although tight. Skipping off small sprocket in front under load. Is there an adjustment possible or do I need a new front sprocket?
The small chainring in your picture looks fairly worn - not bad but not good.
I can't see much of the big chainring to make a judgement there.
Here's a silhouette of a sector of new chainring, with this one piece mirrored left-for-right.
Compare this with yours:
The top photo has a definite "up and to the right" look while the flipped second image has a "up and to the left" look. its not a straight-on photo but you get the idea.
As a test, you can try flipping the chainring, but this will mess up the horizontal offset so can't really be ridden like this with indexed gears. If you have friction gears this may work. Also the chainring bolts may not seat very well against the other side - this is only a suggestion to help test the wear of the chainring teeth.