Weight to one side won't really effect the steering because you naturally shift your body to compensate, it's called countersteering....
It's important to maintain the maximum rated tire pressure (on the sidewall) cold inflation). Bike tires will readily cup if underinflated.
If you ride mostly freeway, there is a good chance the steering head bearings have flat spotted, they need to be checked for free movement and proper pre-load (detailed in the shop manual)
Triumph thoughtfully (cheaply) makes it impossible to lube the headstock bearings, much like draining the forks. It's a simple matter to drill the headstock for Alemite fittings and keep the headstock greased. That eliminates any bearing problems from water ingress (when you wash it) and lack of grease (Triumph uses a scant amount in the bearing cavity anyway).
The forks are another story. I machined the lowers on my bike and added drain plugs but that entails disassembly and machine tools.
I've actually added drain plugs to all my bikes (forks). The USD forks are a bit harder to fit drains to, but doable.
If the bike veers to the side riding hands off, it's mostly the headstock bearings, either preload, flat spotted or rough from corrosion internally.
If you have a set of pads dragging, it's easy to determine. Just ride the bike for a few miles and coast to a stop. Feel the rotors (each side) with your hand. They should be just above ambient temperature (the pads drag slightly, it's an inherent design). If one rotor is hot, that side is dragging. Time to bleed the brakes (in the shop manual again) and check the pads for glazing or uneven wear and yes, I have pads on site for your Rocket....(shameless plug...
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