My 2p...
First of all,
none of Tascam's 4-track cassette-based recorder/reproducers have phantom power, whether they are a stand-alone deck (i.e. 234, 134) or a Portastudio all-in-one unit (i.e. 144, 244, 246, 414, 424, 464, 644...all the "Porta" models, etc.)
If you go with an older one (like for instance the 244) you'll get a more robust chassis and (arguably) electronics, and a greater potential for some TLC. If you go with a newer one (i.e. 424, and I'd recommend the mkII or mkIII) you'll get some very usable bells and whistles (2 locate points, auto-punch, rehearse, etc.) and the logic transport to go along with it...they are also setup nicely to function in a MIDI sync environment with an external sync box if you are into that sort of thing (exceptiion being the 644 which needs nothing external...its got it all).
As for the sound?? Hmmm...I'll let others chime in on that. I've got experience with the 488 and the 424...I'm soon going to be in possession of a 234, but I really can't offer a subjective opinion about the sound quality between early and late generation units. I've heard really great stuff off of both, so YMMV for sure. Neither is going to be the magic bullet. Its going to be (here comes the broken record) dependent on the performance and what's in front of the recorder/reproducer.
I'm not sure about the "graininess" that you mention. In the early 90's I was in a band and we did an entire project on a 488...never had any issues with what I would call graininess...we had problems with flutter but it was a problem with that particular deck (which we were renting). We swapped it out and the issue was better. I believe the preamps are the same between the 488 and the 424, as well as much of the mixer section electronics, so if the dissatisfaction you are having is stemming from that you're not going to escape it with a 424, but...what mics are you using? What are you recording? Are you using external processors?
The 4-track units, with double the tape real estate, have the
potential to capture dynamics and bottom end signals better. I say "potential" because, again, it all depends on what is going into the unit and who's twisting knobs and punching buttons. My experience with the 488 has never left me...I was surprised at the warm natural sound for such a narrow track width (I also have some limited experience with the 238 which, after being away from analog for years and years and working only with digital I was *floored* at the warmth...I didn't know what to do with it all...sounds melodramatic I'm sure, but that truly was my reaction)...the 8-track units...really a marvel and far beyond what Philips ever dreamed would be done with their little cassette.

You'll get more of the same with the 4-track units.
If I were you, I'd see if the graininess is possibly related to the signal path aside from the 488 at present and keep the 488. The mixing section is better than the 424 in terms of flexibility, you've got inserts and phantom power on inputs 1 & 2, not mention the flexibility of more tracks...unless you don't need 8 tracks and you are wanting to simplify...you'll get a lot of support for that here.
Does that help at all?