I find 4K7 pull-up's good from one device on the I2C Bus, right up to about 8 devices on the I2C Bus. Have you put a scope on the Clock and Data lines to see how clean and square the pulses are?

If you are putting your hand over the product and it affects how it works, looks like a decoupling/noise problem. Your I2CBus is pretty resilient and should not be affected. You may need at add some 100nF's around your Vdd and perhaps improve the grounding. It is possible the problem is not with your display, but with your controller. The 18F's are a lot more intollerant of noise. Another thing to look at is if you are using a crystal, if it's being used in weak XT mode instead of HS mode (higher level of drive), your hand approaching it may just add enough extra capacitance to stop it from working.

PS... message me off-list, I'd like to know who you use to manufacture in Poland, what they're build quality is like, and how cost effective they are to use. Obviously the labour is cheaper than Germany, but how much cheaper? Always looking for good intelligence on where to get things made cheaper...