A common issue that's pretty easy to diagnose happens when the pegbox isn't carved deep enough and the string is wedged between the peg and the pegbox below. The A string won't stay in tune or won't tune up as a result. It might feel like the peg is slipping, but if you take out the string and notice that the winding at the end is fuzzy or fraying, there's a sign it's rubbing on the wood below.
The solution is to use a small gouge to cut the pegbox deeper near the scroll, then use some black fingerboard dye to cover your fresh cuts in the wood. Restring to see if it's deep enough, but be ready to unwind and keep carving out the box until the string no longer makes contact.