Use both outlets or just one?

Loc what you're doing is fine; you can use both outlets with no problem.
I suggest (if you're not going to install a dedicated circuit, and BTW you only need one, not multiples) then you might try upgrading just the outlet itself. Many options available in that regard.
I have the Wattgate 381. Expensive but worth it, at least it was for me ... Great results, even over the Hubbles or Levitons industrial grades. YMMV depending of course on the level of your rig.

Why spend thousand(s) on audio gear and nothing on source wiring? Hire an electrician for a few hours and have some dedicated outlets run to their own breaker(s).

many amp manuf.suggest plugging the amp directly into the wall and the front end into a p/c. If your p/c has surge suppression, you are losing that as to your amp ( won't protect you from lightning anyhow)but some p/c will clip power from the amp so thats the trade. Some amps (mac)supposedly have built in protection so not losing anything there.As an example, Richard gray p/c claims their p/c's don't clip and p/c protection inures to both outlets when plugged into one so you could go all into their p/c.