I believe you have enough evidence to know what to look for ( high resistance in a device or connection) and where to begin( the dirty fuse box). A contact/connection with high resistance overheats in use causing resistance to increase which in turn results in increased amps. You need a meter capable of measuring at least 20 amp and preferably fused in case of overload. With system cold/not been in operation for a few hours, start engine then test across problem fuse for voltage drop. Any amount,even a fraction of a volt is cause for concern and likely the problem you are looking for. Remove all fuses and closely examine fuse holder #3 for corrosion and sign of discoloration from being hot. Whether you use Breakclean or soap and water to clean fuse box I recommend using electrical cleaner afterwards to assure contact surfaces are clean. In case of sever damage Scotch Brite pad or 400 grit sandpaper might be required. I can attest that Deoxit D-5 works and Deoxit Shield S-Series keeps it clean for years.
If there is no voltage drop across fuse,kill engine,remove fuse #3 and put amp meter in it's place. Restart engine and note amps. If amps are 13 or above I would suggest moving part of load to an empty slot or in-line fuse. A load of 75% fuse rating is about right. Once fuse slot #3 is 11 amp load or less,keep meter in place of fuse and watch for amps to gradually increase over 20 minutes or suddenly go up sharply. If amps slowly go up but remain below 15 amps,stop,reinstall 15 amp fuse,restart engine and test across relays,wire plugs and other connections for the slightest voltage drop. I have a dedicated pair of meter leads with 1/8" long sewing needles soldered on tips that alow me to penitrate wire insulation without damaging. You can poke straight pins in wires then touch with meter probes.
If amps shoot up suddenly that might indicate short circuit or shorting device. .If that happens we can discuss ways to track it down.
Good luck,if any of this is unfamiliar there are some folks here that can answer questions and/or suggest different approach.