OK, mini-lecture on electricity: My hope is that it will help a little when non- technicians are trying to analyze and trouble shoot simple electrical systems.
A very good analogy is to compare the electrical system to a water (or any hydraulic) system. Voltage is pressure, current (Amperes) is flow, wiring are the pipes, switches are valves and loads like lights and motors resist the flow and pressure develops across them.
When a load is wired in parallel like headlights, they all get full voltage and the amount of current available is divided between them just like splitting two hoses of off the main spigot. If one of them has some resistance like a crimp in the hose (rusty or otherwise bad connection) some of the pressure (voltage) is shared with the unwanted resistance, hence your pressure is lower at the end of one hose; in your case the light is dimmer. If a wire or connection is too small it is like trying to squeeze water through a small opening so again, lower pressure or voltage. That is why high current devices need large wires and solid clean connections.
Relays allow a small switch and wire to control a large current (a small electro-magnet attracts a steel plate with larger electrical contacts on them). If they have bad contacts or connections, you get current flow problems. If the relay is pulling in (and staying in) you need not check the switch and wiring that controls it.
A fuse is purposefully made to be the weakest link. They are designed to allow only a specified amount of current to go through them before they melt. If you are blowing fuses you have too much load for the design of the circuit (either a short circuit or someone added something). That is why you should not put a larger fuse in unless you fully understand how to size them. They are there to keep another (more expensive) part in the system from melting.
So, any connection can give an opportunity for oxidation to sneak in lowering your voltage at the load (headlight in this case). This includes ground wires which are often overlooked (this is the return path to the battery). In addition to checking fuse connectors, relays, sockets and the like, check that the frame and body panels have good ground connections as well.
Whenever possible use emery cloth (available at most automotive and hardware stores) instead of sandpaper, as it will leave a better finish and is not as likely to leave particles on contacts.
Good hunting,
RDB