Usually a relay is there to protect the initial supply switch from too much current arcing as it closes. The initial switch only has to deal with the lower current to supply the relay coil, and then the activated coil pulls a heavier duty contact (switch) together to handle the heavier current of the end load (in this case the headlight).
If you have no relay, your intial switch could be burning out or has arcing damage causing a voltage drop across itself and since it is in series with the rest of the circuit thereby increasing overall circuit amperage. Check your switch that feeds the circuit for such damage.
Your really need a schematic to sort this out, it will give the color and trace of everything, otherwise it's a nightmare physically tracing each wire to it's end. On my 2011 Nomad schematic a wire coming from the ignition switch feeds thru the fuse and then to the headlight relay. If you can find where the relay "used" to be, then trace that wire back to the fuse. Is that the fuse that's blowing ?
Additionally I surmise someone has bypassed the relay with this fused feeder wire and hooked it straight into the line feeding the HI/LO switch (which normally comes out of the relay). Both the ignition switch and the HI/LO switch could suffer arcing damage without the relay in place.
A test on each switch with an volt meter "should" show the same volts leaving the switches as what's going in them. Otherwise the switches have damage and are dropping volts and building series current amps. I don't get online as much as I used to so I may not get back to this thread for awhile. But the other guys can help you along after you test both these switches.
There also as said the possibility that the wiring has been foobarred, but with the relay "gone" I would first check those switches for damage.