Chain overshoots cog when shifting from low to high gears
The following links will show you how to adjust a derailleur: 

http://sheldonbrown.com/derailer-adjustment.html

http://www.parktool.com/blog/repair-help/rear-derailler-adjustments-derailleur

Note that cables and derailleurs do need replacing from time to time and they do need occasional adjustment, but try to follow one of the derailleur adjustment systems suggested in the prior links (assuming the derailleur isn't bent or something - if it is, take it to your LBS to be straightened). Limit screws only keep the chain going too far into the spokes or out of the cassette, and do not affect shifts between the largest and smallest cogs (only the extreme points of chain movement), so those should not have been tweaked (probably). 

Note that bending a derailleur is quite easy even with a small spill. Your LBS probably can adjust this in 5 minutes if you can't figure it out.