Rear derailleur low limit adjustment changes under load I have a hybrid bike, which, when I am sat on the bike, shifts and remains into the largest cog on the rear cassette with no problems.

However, if I take my weight off the bike, then when I start pedalling again, the chain skips between the largest and second largest cog until it settles back on the largest cog. Also, it is impossible to get it to shift into the largest cog when the bike is upside down. 

I've tried loosening the L screw, but it hasn't worked yet. I'm guessing that if I keep loosening it it will eventually work, but the fact that the adjustment is so dependent on the load on the bike is making me think that something else might be wrong. Is there anything to worry about out or should I just go ahead and keep loosening the L screw?
Before you go any further, you should check (or have it checked at your LBS) your derailleur hanger. Oftentimes when the RD acts up and won't "tune", it has a poor reference point to start from. The derailleur hanger is easily bent out of alignment,and can render the rear an unreliable little beast. 
Double-check for any side-play in the crank, to see if that isn't a contributing factor, and get that hanger looked at. Good luck.