@bound4h I’d be in the camp that if you aren’t currently happy with your speakers changing the amplification isn’t likely to cause you to fall in love with them. Amps will amplify no pun intended strengths or weaknesses of a speaker but it won’t change the overall sound signature of the speaker. I’d be looking for new speakers prior to an amp. I had a set of Focal Electra 1028BE’s highly regarded as great value, trickle down effect on tweeter / woofer technology in Focal’s more expensive offerings. Some songs, music were breathtaking but overall the sound signature was too forward, bright. Most of the music I wanted to listen to wasn’t as enjoyable as I wanted the experience to be. I chased Amps, SS, Tube with the lean toward warmer sounding amps to tame the Focal’s. In the end, got some sage advice and tried other speakers, focusing on speakers that had the sound signature I leaned toward.
If you go the route of a new integrated and same speakers - Coda CSiB if you can find one used will be under budget and it’s a beast - the pre and amp sections are fantastic - it drive just about any speaker. Krell 300i is another fantastic unit but likely over budget even used. The Music Fidelity Integrated’s usually get high marks for punching above their weight class.
good luck!