Square rooms are rough. Depending on what you've done for set up, i'd explore keeping that sub for now but making certain you have it placed in the best spot in your room! If you haven't, experiment with the Subwoofer Crawl, first. There's a video here, as well as others you can find with a quick google search.
First, keep your mind, and your ears open when you do it! I learned a lot about my room acoustics and low frequency soundwaves while I did it. I tried with pink noise, test tones, and a really good bass track... it took me several crawls around the room before I started to hear it all, but you will. I had a corner that didn't just amplify the sound (premise behind corner loading) but it also got super boomy and muddy sounding. Not there! I found a spot on the back wall where the bass just flat out died. Not there! Fortunately, my side walls had long stretches where I could consider placement that allowed the bass to sound like I wanted. Clear and real. Some people will say that in just the right spot, you'll get an extra punch and tightness out of it.
You do not need to put it on your seat, just move your seat out of the way a little if possible and put the sub in your LP. You can put it in your seat... either way, won't hurt anything but maybe your fine leather chair.
Try to find 2 or three spots where the bass sounds good and mark them with tape. Switch the sub to that position, go back to your LP and see how it sounds. It should sound the same!
Regardless, it a crude way, but it might help the problem a little.
I said keep your mind open, because you might find the best place is right behind or beside your LP.
Beyond that, How tall is your ceiling? And is your room closed off or open to other areas? Low frequencies work on room volume rather than distance, so if that 10x10 room is open to another area, your dealing with a much larger volume than you realize.
Fotgot to add... the reason for multiple subs... especially in a square room, is that the way the soundwaves move, you start to get cancelations... voids, nodes, deadspots. Multiple subs, placed smartly, will work to even that out and give you better response.