can a new amp amp tame my speakers?

IF I change speakers, and it's a big if, I'd be more likely to have to get some that are more streamlined and 'fit' the room better aesthetically. The PSB's are 'approved' in that regard. Having said that, I've listened to B&W CM9's (I think that's the model, about $3K?), Paradigms in the same price range, Studio something's, neither of which impressed much, although a slight edge to the B&W. I did hear the 804's, and was impressed, but not in a particularly great listening setup. I heard the Vandersteen Treo (again, I think, about $6K) that someone was using to audition a TT. Associated electronics were also significantly upmarket on those, but they did sound pretty good!

I haven't covered all bases by any means, but so far what I've heard is leading me to believe that I'll need to get close to $6K to get to an improvement in sound that's more than just incremental and doesn't involve a compromise elsewhere - although I am also aware there would be a 'boost' to the sound of most of what I've heard by getting them into my room vs. the setups they've been in, but that applies to the PSB's as well. That I'm not ready for. I would be quite comfortable buying used speakers from that level though. I do see a set of Synchrony 2's on the 'gon for a great price, about 1/2 of list, and they are really tempting...
...but again, how good would they sound with the amp I have, a 25 year old Yamaha that was nothing special when I got it, never mind the march of time, although it's never missed a lick in all that time either...
What I'm looking at mostly is amps in the sub-$1500 range. I can't help thinking a good quality amp will bring the most out of the speakers I have, and when it is time to upgrade the speakers, I'll have quality to drive them with already.

Other items in your system besides amps can change your sound, I would try some different IC's. Try some old monster cables. I remember I did not and still do not like monster cable because to me they seem too polite especially on the high end range. It could also be a lot less expensive depending on the IC you purchase. Go for IC's that are on the warm side of neutral.

Hmm..I have Audioquest from the AVR to the amp, which was an upgrade over the 'Radio Shack' level I'd been using, but heonestly, I didn't notice much if any change.

I have owned Audioquest and found them to be more to the lean side of neutral. They could be why your system needs to be tamed. Have you tried your old IC's since your speaker change, They may sound good compared to the new IC's.
I would look for IC's that are some what on the dark side of neutral. MIT may work well for your system but can expensive unless you get some used or try monster. Also remember new IC's do need time to breakin around 200 hours and will sound very different over time.

I think your idea of a good amp to mate with your speakers is a good plan. Then fine tune with IC's as Hevac1 says. Then move to better speaker if needed.

If you buy used you can keep $ losses to a minimum if you don't like the results. Cables can be changed like underwear with this theory. Amps and speakers get a little tougher.

Btw, almost any of the amps mentioned would sound good. Be sure and check input impedance of your amp against Preamp impedance.