Not sure if you can hotwire cut foam with a 3 axis machine. Mine has 4 axis with 2 on the left and 2 on the right. Two are x axis and they are all independant from each other along with the two y's which are towers for the vertical movements. With yours you probably would still be using the router as I don't know where you would connect the wire.
Mine is a Hobbycnc pro 4 axis board which is reasonable and I use drawer slides(ball type) with ready rod. It is a combination of what I have seen and I am quite happy.
If you have any questions I would be glad to answer with my limited knowledge.