Is it time for a new socket set?

Just a general wrenching question here.

I went to tighten the chain on my 2022 SV650, and while I can spin the bolt when rotating one of the axle nuts, I cannot actually loosen it for the life of me. It's 19mm on one side and 27mm on the other. My 19mm socket just will not grip, and I don't want to keep trying because if I do it too much more I'll round the nut off for sure.

I don't have a garage at the moment, and I don't have air tools, just hand tools.

Is rounding nuts a sign that my sockets are getting too worn and need replacing? They have served me well for like 20 years, but I don't know if they wear, how quickly, etc.

Thanks!