What if your charged with a dui and the officer doesn't give you citation papers but wrong copies of different papers when your released do you then still have to go to court and does this fall in statue of limitations after time?

No, once formal charges are made with the court the SOL stops. But even so, it would be a lot longer than a court date.

What happens if you don't show up for court, they put a warrant out for your arrest for not appearing. Then after you are in jail, you will have to try and convince a judge you did not get the right papers. ( most likely they will never believe you)

First, WHY were you driving drunk?? Second, you need to take the bull by the horns and do whatever it takes to get the mess cleaned up.

FrChuck is right - appear as scheduled (call the Court if you have to in order to determine the correct date/time) and make your argument there that you received the "wrong" papers. If you did, you will then be given the "right" papers. If you fail to appear a warrant (which never dies) will be issued for you.