Is it possible to buy a decent tube amplifier in the $2000-3000 range?

Imo get rid of the speakers. They are not efficient; they are like pushing a brick, like a Magnepan. It would take a crapload of power/current to make them sound lively. By looking for an inexpensive tube amp you are setting yourself up for marginal improvement. They are, of course, lovely in terms of nostalgia, but there are far, far better speakers now. You can do so much better in terms of speakers.

Don't fool yourself into thinking the speakers are "all that". If you want a sea change, change the speakers. Or, rip into them and redo them, from crossover to internal wiring to drivers. If you love their appearance that would be worth it to you. But it won't help with valuation of the speakers. 

Don't expect miracles if all you do is change amps. You would get much farther overall by changing speakers, imo. Your current electronics with a more efficient speaker would be revelatory to you. :) 

+1  Change the speakers.

Music Reference RM-200 Mk.2. Occasionally available used for around $3000. Or an RM-9 Mk.2, for around $2000. The RM-200 is a tube amp unusually good for use with low impedance speakers, due to it's lower-than-normal output impedance and increasing power with descending speaker impedance.

With barely "efficient" speakers (allegedly 91db) my $1200 (1500 new when available) Dennis Had Fire Bottle SEP HO amp, designed and hand built by a genuine American geezer (only mentioned because hand wired SE amps from some places cost multiples of this…Shindo anybody?), is one of the best sounding amps I’ve ever heard and produces some genuine loudness with its mighty 12 watts per side (or 17 watts…not certain)…a world class amp which made me reconsider my opinions regarding wattage and geezers.

I also recommend Primaluna. You can frequently find used product online. I would look for the dialogue series over prologue. I think echo audio in Portland had a great deal on some mono blocks. You could buy one and use in stereo or buy both. They do sell online but don’t always ship depending on the product. I have a dialogue one integrated and I love it.