Don't you love the neverending sprint?

You know the one that just inherits all the tickets from the last one? Each sprint packs on new cool features but along the way bug tickets come in for code you haven't even touched or wrote and you deal with them because they are blockers.

before you know it, sprint is over and ok to the next one. I'm pretty sure that is how this is supposed to work right?

How about keeping track of new features and blockers? And show that to your (project) manager. Try to stay away from complaining just present it as a fact.
You are spending your precious time on these blockers to avoid the nasty situation of not being able to deliver/deploy at Sprint's End.
Let them know.
The best project manager i worked with had 30% of the time booked as 'unforeseen', perhaps thats and idea if these blockers keep showing up.