"After 20-30 minutes use the computer locks up"
could be heat related but i doubt it.
"It would appear that while running in safe mode, everything is OK"
if it runs ok in safe mode then this would suggest software.
click start then run and type msconfig and click ok.
click on the start up tab and any program you recognise untick the box to stop them automatically loading with windows.click here for reference if you are unsure of the name.keep norton ticked as you need this.
click apply and ok when done and restart.ignore the warning after restart and click dont show again.
you can also disable uneeded services click here for a guide,this will help free up some memory.
have you tried running the disk check in safe mode?.this will normally work.