Yes, axe is correct in American English, and is actually the more common spelling. However, some style guides require that it be spelled ax, so it is important to check.
And remember, being consistent is also important. Don’t switch between ax and axe. Choose one and stick to it.
Axe is a noun that describes a long-handled tool with a sharpened, wedge-shaped cutting head used to chop things (typically wood).
It is also a regular verb, describing the action of chopping with the eponymous tool.
To make sure you have the right spelling of ax or axe, you can use the QuillBot Grammar Checker.