Badger pretty well nailed it! You could try Youtube for cutting veneers on a bandsaw as well. Sounds like you are well on your way though! less teeth / inch is definitely the way to go. Too many teeth will have small gullets leading to sawdust clogging , overheating , and blade drift. The coarser teeth shouldn't give you a rough cut on rips. If it does ,it is likely because a tooth or two are set (bent for kerf) a little more than the rest .This will usually happen right at the weld. Find it and straighten it a bit if you are worried about it. With a good set up , even a modest (not crap) saw should give you good results. Try it on scrap first ,as always. Good luck!
BTW , the "veneer " blade for the table saw is for cutting veneered plywood, etc. No good fro making veneer. Too many teeth will clog the cut , burn the wood, warp it and your blade etc. etc.You need a good ripping blade with 24 teeth / inch no wobble and a zero clearance plate. A high quality saw is key here though. With this setup on my saw I can dial in very accurate thicknesses that can be glued up without resurfacing ( it's that smooth) . Constant feed rate and adequate support are necessary components as well. You can also cut a pair of tapered lams if you need with this set up quite easily.
I would still say the bandsaw is your best bet!
