It is worth pointing out that amplification is never only about just making an instrument louder, some tone-shaping is always involved. If you were able to take the sound of your '22 Gibson and ramp it up to rock band volume, you would probably find the result quite unpleasant, if not painful - very hard and penetrating; the solution is to reduce the high mids (it has to do with how our ears "hear" sound - that's what the "loudness" button on old-fashioned hi fi amplifiers was about).
The goal is to produce a signal that is loud enough to be heard in an electric band context that still sounds like a mandolin and is pleasant to listen to. You write that you were unhappy with an Eastman because it "sounded like an electric guitar" - the fault may well not lie in the instrument, rather in the amplifier - there are no commercially available electric mandolin amplifiers, and most people end up using guitar amps which are voiced to produce an electric guitar sound, as you discovered. It is possible to modify a (valve-) amp to produce a more mandolin sound, but not easy to find a technician who can perform the operation (I did it myself by trial and error, but do not regard myself as qualified to give sound advice). Also, the pickup may not be ideal (I've never played an Eastman electric, so I'm speculating here). http://www.almuse.co.uk/mandolin_pickups.html offers specifically mandolin-voiced pickups, which are a great improvement on the standard pickups on Fender and Epiphone electrics (here I am speaking from experience)
Any solution which involves sticking a transducer, be it piezo or contact mic, on the soundboard of an acoustic instrument has its limits as regards volume, although I believe there are some very effective feedback killers (in effect, intelligent notch filters) available. Playing into a microphone, be it clip on or on a stand is more prone to feedback than the pickup route.
For my acoustic mandolins (1908 Gibson A3 and a Vanden from '86) I use an AKG c411 contact mic direct into the soundboard, and if it gets really loud or I'm just feeling raucous, I plug in a custom-built 8 string solidbody with an Almuse "Aggressor" pickup into a 15W valve amp that I put together in the '90s.
Good Luck!
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