While I own and use many (many!) axes, hatchets, and mauls, I am always cautious about inexperienced users. An accident from an axe or hatchet can be very serious.

Also note that an axe with a thin "felling" profile (e.g., Estwing) makes IME a lousy and potentially dangerous wood splitting tool. The cutting head can easily become deeply embedded in the wood without splitting it, requiring all sorts of contortions to get it out.

A good folding or bow saw is a very practical choice for a self-propelled traveller. If you can't split the wood, you can cut rounds off the end that will burn reasonably well.

However, if you split the wood it will certainly burn better. Consider this: the primary hazard with an axe/hatchet comes from swinging it at a target. But you can also use an axe or stout hatchet as a simple splitting wedge and "baton" it with another piece of wood. Much safer. If the wood does not split readily (too many knots), use the saw to cut it into shorter lengths that are easier to split.

And last, a dumb question: have you determined if there is already a splitting axe at the hut?