Crossway is another game that uses a Go board and stones. This is essentially Hex on a square grid. The issue of diagonal paths crossing is resolved by forbidding either 2x2 pattern of opposing paths crossing from occurring anywhere on the board. Blocking is difficult but not impossible. Game play can be brutal, more like Twixt than Hex.
I would also like to mention that Go on a grid of triangles could be categorized as a Go variant that avoids Komi. No snap back can occur on such a grid.