Not to worry. Chances are your 5 gal carboy is actually larger like everyone else's.
As long as you have the airlock on you can't get oxidation. The CO2 coming from the brew is not allowing fresh air (and oxidation) from coming in.
Another thing, next time your carboy is empty take a gallon jug, fill it with water and pour it into your carboy. Now get your girlfriend/wife's red nail polish and put a 2" line on the glass to mark the 1 gal mark. Repeat the process for 2, 3, 4, 5, and 6 gals.
Do the same to your plastic bucket, ESPECIALLY your bottling bucket.
These lines will let you know how much brew you REALLY have when you get ready to prime for bottling.
You really/probably don't want to put 1.5 C of priming sugar into 4 gals of beer unless you want a lot of carbonation. Adversely, you don't want to "guess" at using 3/4 C for 5.5 gals.
Happy brewing!