your rear wheel trouble is because the rear is 'out of dish' to one side, you need to re-dish it and if your rear wheel fits into your bike very securely and always in the same position exactly you can make a pretty decent job of it by removing the tire and re-dishing it on the bike... it is most likely pulled to the non-drive side (the side without the gears) and you will need to: loosen all the non-drive side spokes evenly or tighten all the drive side spokes evenly or a combination of both... if it is an old wheel this can be difficult, a drop of oil in the end of each spoke at the nipple can make things easier... when dishing a wheel an 1/8 of a turn on each spoke is a reasonable starting point, if you have trouble guaging an 1/8 of a turn you can try a 1/4 turn to start... its easy to get lost when adding tension to only one side of the wheel so its best to start at the valve hole and tighten or loosen every other spoke until you get back to the vavle hole again...
with the tire off it is easier to see if the wheel is properly dished (back to center) you will start by putting the wheel in the bike normally and then removing it and putting it in the opposite way, this will often make it easy to spot an even slightly out of dish wheel, as long as the wheel is in the same location both ways... look at the rim in the frame between the brake pads AND between the chainstays behind the bottom bracket, one other problem to look out for is a bent axle when trying to re-dish a wheel, it will move the rim around depending on which direction the bend in the axle is facing...
this is a somewhat advanced operation, but if you have trued your wheel and have ridden it a while and it is still staying true you are probably ready to give it a try...
all's'miles
curby