Not quite correct there Dave. Unless you have a constant power load as distinct from purely resistive as your bulbs are, any unintended extra resistance in a circuit will actually reduce current as per Ohms law I=E/R. What it can do though is produce a volt drop at point of localised resistance as with a bad connection, translating as power dissipated as heat. This explains why images of melted connectors appear regularly on this forum.

Your fuse issue was because it was working for a substantial period of time at close to its maximum current though, as well as likely having slow blow characteristics. Consequently, and this may seem ironic, the poor contact you found on bulb holder may have reduced current just enough to stop the fuse blowing.

The answer would have been uprating the fuse value as originally suggested which would have stopped the fuse overheating whilst providing a tad more power to your bulbs. However your decision to go with LED equivalents sounds a good call.

FWIW my car is a 100% LED illumination fit now. I even reduced fuse values on the side and headlamp circuits to provide as much protection as possible given reduced load current.