OP may have a hard time figuring out what's what, given he (?) thought the spring in the QR had something to do with clamping the wheel...
But anyway, a bit of persistence and applied nous should get you there... for a start, I doubt you need new QRs; you've prolly solved the issue already. I find it's usually about right if the QR starts to bite when the lever's halfway home, ie in line with the axle. +1 if your cables aren't kinked or frayed, you can prolly get away without replacing them, just clean and lube them.
However, it sounds like your chain and sprockets may be worn... if you still get the chain jumping issue after you have the indexing dialled in (so the derailleur's upper pulley sits directly below each cog you shift it to), then it's prolly worn. You can measure 24 (tensioned) links to see if it's much over 12"; any more than 1/8" over means it's toast. Unfortunately, it'll cost you a bit to replace the chain and cogs (prolly something like 30 quid)... you can avoid the chain skipping somewhat by spinning faster in a lower gear.