Sometimes on a PCB like that, where you have the long term heat of resistors baking it, it will not only foul the joint but also delaminate the copper foil off the PCB and it can then oxidize away or tear. For this reason it can be useful, and still might be repairable by soldering again, to follow the traces to their next destination on the PCB and solder on a jumper wire between the resistor leads and that next solder joint at the next destination. Because a jumper wire can flop around, I would use some cement or epoxy, or sensor-safe RTV, to glue it to the PCB to make it more secure... after making sure it works again, and trying to keep the *glue* off any other solder joints that might need the same done in the future if each lamp subcircuit fails one after the other, eventually, though the headlights are the more failure prone because they use the most current by far, creating more heat on the respective resistors for them.
However, yes you can be rid of the LOM. Besides the data wires that trigger the signal that a lamp is out, it's just sitting in series with the power feed to the lamps going into it (measuring that across those troublesome resistors), then the power goes back out to the lamps. To get it out of the circuit for the right headlight you would just solder together the dark-blue/light-green wire, to the dark-green/orange wire.
If you want to do the same for the left headlight then solder the white/light-green, to the dark-blue/white wire. I have a second wiring diagram that contradicts the wiring to the left headlamp and suggests connecting the dark-blue/white wire to the same dark-blue/light green wire as on the right headlamp, which should definitely work if connecting it to the white/light-green doesn't work or that wire isn't present.
Because of the way the module is set up to work, this will cause your dash indicator to still indicate that the lamps are out if you leave the module plugged in, but if you unplug it, then it will break the circuit powering the rear lights, parking lights and brake lights. There are then two options, either solder the input and output wires for the rear lights together as done for the headlights, -OR- you can sever the signal wire that tells the dash module that the lights are out (if that is unsightly to see) and leave the connector with the remaining wires plugged into it.
The wires to sever would be as shown on the attached wiring diagram.
I don't know all the module years but this was usually a higher end feature so it may be on all 2nd gen Limiteds and Eddie Bauers, or at least that's the place to look first for any from '95 to '01 and possibly even past '01, but be aware that this is a common problem that happens to all of them eventually from heat and time, so you might be as well off repairing what you have or bypassing it rather than having to do either of these things to a module from a junkyard sooner if not later.
If your driveshaft bolts don't look torn up, I'd make sure to torque them to spec and if you did that previously, locktite should help.
