I find with mine that the mouthpiece has to go hard down over the cork to get it in tune, further than with the alto. Plastic reeds are great. I use a Legere on clarinet - not cheap, but it's consistent for the short space of time I'm using it. I have heard, however, that plastic reeds can soften a litt as they warm up and this throws the tuning out.
You can certainly change the pitch by trimming and clipping cane reeds, but I know little about how to do it, and I've never heard of it being done with plastic ones. I'm off to band practice in a a little while, so I'll ask one of the guys there who runs the sax department in our local music shop.
If you have problems with drying out, invest a few quid in one of those reed holders with a lid on them and a little tube of 'vitalizer' integrated into the design. I often take reeds out of mine a day or so after playing to find the reed still damp.
What reeds have you tried? I'm currently using Vandoren ZZ on a BG plastic mouthpiece on the soprano, and on a Lawton mouthpiece on Alto. They play pretty much straight out the box, and I find Vandoren a good, consistent reed, although I know plenty of players who swear by various other brands.