I used to have this exact same issue 4 years ago. Solution was to remove the wheels. Depending on your chair, you might need a screw driver, but either way it’s really easy and shouldn’t take more than 5 minutes. Just get rid of the wheels.
Can we make a little suggestion which may help with this instability issue.
How’s about increasing your immersion in our beloved Sim by having some Chocks on your Wheels attached to a handy Rope.
When it’s time to go and make Coffee or feed the Cat you could yell CHOCK’s AWAY ! and become mobile.
ps. If you have a Cat or a Dog it wiil surely know that it’s time for Din Din’s.
There are suitable small Chocks in this selection that may do the Job.
On Amazon real cheap: Bell Glides Replacement Office Chair or Stool Swivel Caster Wheels to Fixed Stationary Castors, Low Profile Bell Glides with Soft Rubber Bottom Instead of Self Felt Pads.
I bought those too but everytime I got up from the chair the casters will get off the pad and it was a pain to make all five caster fit in the little cavity (I have a heavy chair). I did find a use for those: glued them to the bottom of my Thrustmaster Rudder Pedals and they no longer move on my wood floors.
I have these for my chair:
I’ve been driving, and now flying, with my G27 pedals which have a brake which is very stiff indeed (the pedals are braced to stop them moving). My wheeled chair stays pretty well in place as I have always had it sat on one of those commercial ‘dust’ mats: