With load cells you are working with low voltages. Unfortunately, with such low voltages, thermal effects come into play - dissimilar metals (wiring,terminals) for thermocouples which give you thermal drift. Also with op-amps they too are thermally sensitive therefore there a chopper-stabilised op-amps that alternate the input with a chopper to cancel the offset effects. These are more expensive. Also, noise becomes a major issue of which you're amplifying with the op-amp. This gets fed into the a/d and quantitised. If all you're getting is +/- 1 digit drift then that is pretty good. Browse the web for app notes and articles on strain gauge conversion. Texas Instruments have a good app note on using their MSC1210 for weigh scale applications where they go into noise and errors. You've stepped into a fairly specialised area - it might be easier for you to buy equipment suited to your task - it's not cheap for a reason!

I'm working on a similar thing at the moment and I use a 6 1/2 digit HP meter for measurements - I'd have no hope with a standard meter!