A couple of things leap to mind, it could be dirty contacts on the ROMs, or between the ROM board and the main board. Worst case scenario is that you have a bad SRAM chip on the mainboard, or a blown ROM, however it looks very much like an address bus problem, the simplest cause for that would be that a jumper has fallen off the ROM board.
The ROM carrier boards were not game specific so they used a lot of jumpers to set the board up for the game that ended up on there. If one is missing or in the wrong place you end with weird address bus issues affecting some or all of the ROMs.
Assuming your GA has the same ROM carrier board as mine then the jumpers should be as follows
Golden Axe
=======
s1 No
s2 Yes
s12 No
s11 Yes
s10 No
s9 Yes
s18 No
s17 Yes
s14 No
s13 Yes
s4 No
s3 Yes
s16 No
s15 Yes
s5 No
s19 Yes
s6 Yes
s7 Yes
s8 No
If your jumpers are wildly different from this pattern then you probably have a different ROM board to me, if you do have a jumper fault your setup should match this very closely, with possibly only one or two missing.
It is also possible that the seller never played the game to the point where the fault showed up.