1) The cheapest without using powerline is to use a multifunction Wireless AP (like this) and convert it to bridge mode. Since it will use wifi to connect to/from the main router, it will not be fast. When you move to your new home, you can convert this to a wired access point for added coverage.

2) MoCA input and Ethernet input? Find a MoCA-enabled router/extender, which is VERY scarce, and is usually offered by ISPs (like Verizon or Frontier).

The solution really is MoCA adapters if you want stable wired internet through coax.