They sell a variety of combo Tube/Solid State amps now, as well as some "modeling" amps that can electronically recreate a lot of warm sounds. However, for true tube warmth, nothing beats a tube. I recommend buying a tube/solid state combo. As for the overall sound not being satisfactory, it may be two things:
1. Epiphone makes fine instruments, but the materials are cheaper than true Les Paul models from Gibson. The problem is that you get a flatter sound on lower grade pickups. This can sometimes be easily remedied by hotrodding the guitar with new pickups….and that may be cheaper than a new amp. If you are not technically inclined, they can sometimes do this at a music store for you.
2. The adjustment of your bridge height can affect the Attack, decay, sustain and release of the instrument, so you might simply want to try getting the guitar tuned for brighter sound.

final recommendations:
Change the strings. They should be replaced every three months or every hundred hours of play, whichever comes first. For a brighter, jazzier sound, use a lighter guage and try to play with a softer hand on the right, and more precision on the left.

As for specific tube/solid state combos, go to something along the lines of a Fender Jazzman or some of the Marshall offerings. I don't know your price range, but I assume it is low. New is ALWAYS better than a used amp, so if you can't afford a brand new one now, ask the store about installment payments. Good luck & keep on playing.

As to better overall sound, try changing the guitar to a true jazz guitar. You sound like you are looking for a semi-hollow body sound, and Epiphone Dots are good for that at a lower cost. An Epi or Gibson Sheraton are also good for this kind of music. Between a new amp & guitar, I think you should be able to get great brand new equipment for about $1400 total or less. That works out to about $115 a month. If you can find a way to fit that in your budget, you can trade up within a year, and then you can sell your old equipment to earn some of it back.