is there any mnemonic for the AXE method?
I think you shouldn't abuse with mnemonic. In this case you should start with this rule of VSPR:

each atom in a molecule will be positioned so that there is minimal
repulsion between the valence electrons of that atom.

Now you should begin to train your three dimensional thinking!
This is the method I've adopted to learn AXE, of course everyone has his own but maybe it could help to you to find yours:
Learn to draw the $AX_nE_0$ series:
This for me was the first step: learn to draw the basic geometry 0 lone pair. This is quite intuitive knowing the previous law. Begin with the $A$ then add the $X$s. Note that the linear module could help you to remember it.

Add 1,2,3 lone pairs series
This is the next step. I've brutally modified the wikipedia table to show how you can easily do this step from the Basic geometry 0 lone pair. Simply begin to substitute $X$ with $E$: one $E$ for 1 lone pair, two for 2 lone pair...
Nomenclature
The last step is to add the nomenclature. First of all you should take in account that the lone pair ($E$) don't appear as a part of the final shape from which the nomenclature is derived. If not considering the $E$ group you obtain a planar molecule you have to add the word planar after the name, if you obtain a pyramidal shape you should use pyramidal after the name of the planar pyramid's base.
So imagine that $E$ groups are invisible and try use this method if you  can't find it only with pyramids try to figure out which of the following figures is more close to the actual shape of the molecule.

The first two are platonic solids  (Octhaedron, and Tetrahedron) then there is a T for a T-shaped, and a bent figure. In the original wikipedia table respective molecule configuration is already rotated to the right position, I think you can do it on your own, with your 3D thinking.
For the last, seesaw shaped, there is a powerful &quot;mem&quot;:

Here is an old photo of two boys in a $SF_4$ molecule.Good ol' times!