Having had a similar situation one of the key things we found was missing (or deficient) from our stories was a well defined (and agreed upon) "Definition of Done" (DoD). If you have well defined "done" criteria then when the story is complete (meets the DoD) then it's complete, of course the trick becomes how to define a good definition of done that's not too vague, ensures good quality, everyone agrees on and is actually doable. For me it's not a once of exercise either, we would continually re-look at it in each retrospective to make sure they were still relevant and made changes as needed. For ideas on setting up a good DoD try Google, there are a lot of good exercises teams can do to define it (like this one).
As to when bugs that slip through get done I'd say they should be fed back in to the Product Backlog and be prioritized as per normal. Bugs that come up during the sprint related to sprint work should be fixed in the sprint (avoid admin / paperwork!). Remember also lots of bugs (or an increasing number of bugs) generally mean there's a problem with quality, maybe there's a bigger issue to deal with (too much pressure to finish perhaps?).
To summarize have a look at getting a solid DoD (that encourages quality over quantity) and continually improve it, slow down and do less better instead of more "with bugs" and when bugs DO appear, feed them back into the product backlog and engage the PO to prioritize it as soon as possible.
As a side note we found that when it came to dealing with lots of bugs, a more lean approach like Kanban worked far better than the 2 weekly iterative approach of our Scrum process.
Hope this helps?