I had a similar problem recently with my 97 tib, and it honestly sounds like you should replace the alternator, obviously since you already know its not keeping the charge and I would also check your alternator fuse! It sounds like based on the problems your experiencing (not starting, loss of power, etc.) you should check your alternator fuse and see if its bolted down completely!! Mine was having the same issues (I replaced the alternator cuz it was bad anyways) but my fuse was not being held in by the bolt all the way so it kept losing the connection! 
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1997 Tiburon FX - HP Mafia Flag Girl
Motegi Rims-Toyo Proxy4 Tires
Clutch, Engine
Alpine CD Player
and almost here---header and intake
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Cuda Kit (evofusion.com), Double Sided Carbon OEM Hood (flyryde.com), Paint Job (custom), Lowering Springs (possibly), Swaybars, Strut bars and much much more!!!
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