Don't feel bad about the whole fuzzy mouse/ huge rat thing.
Now, if you had never offered your snake anything at all to eat since you got it, then yes you'd be starving it. Even if it was smaller food, you were still feeding it.
Be very careful about going too big with prey. While they are quite capable of eating something MUCH bigger than what it looks like they should be able to, I have found through trial and error that feeding a little bit small is better than feeding a little bit big.
I try to feed mine prey that is just big enough that a single prey item will leave a little "full belly" bulge. Not big enough to make them look like they swallowed a football, but not so small you can't tell they ate. Think "Thanksgiving dinner", not "Pie Eating Contest".
Gale