Does having an aunt with uterine cancer make me more likely to have one too?

Endometrial cancer. What you describe alone (one second degree relative) does not increase your risk. A complete family history as well full history and physical is needed to know your risks.
Possibly. There is an inheritence pattern for several cancer types, including ovarian cancer. However, knowing the specific type (endometrioid, serous, brenner, mucinous, etc), would make a definitive comment more meaningful.