The 'LS273 handles the first part of what you want to do (latching the outputs). How do you plan to implement the “only 1 on at a time” behavior?
Now that I think of it, a 74LS154 decoder will do exactly what you need (apart from relay drivers and possibly an inverter). This is a 1 of 16 decoder that will decode a 4-bit input to one of 16 outputs. Can’t believe I didn’t think of that earlier! Should not require any more logic than that single chip.