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It could be a number of things. I have an '88 E-350 RV and the left front caliper was hanging up. It would heat up over time and eventually start grabbing enough that I could feel the pull to the left side. I ended up changing both front calipers.
It's possible the smell you are smelling is the brake material heating up, or it could be something totally different.
In my opinion, I don't think an axle would cause a pull to one side like that but like I tell people, I'm usually wrong.
but you did mention the smell of hot oil. bearing grease can give off that smell. So a check of your bearings and your race is in order. hopefully it is not that and you have not scarred the spindle. I have seem many a trucks in my time servicing them in the military and a majority are 4 by with the front axle assembly hanging up because of lack of lube and being over heated
a smell of hot oil is an indication of a severe problem and should be looked into asap. when dealing with drive componets
you could have a leaking wheel cylinder if you have drums and a leaking caliper......
Before changing it you can give it a quick check. Drive it around and when it starts to pull, go home and jack it up quick and try and spin the wheel. If it's bearings I'd think you hear some bad sounds. If it's brakes it should just make an even sound, if you can even turn it.
You may be able to feel something getting hot to give you some clues, be careful, it may be hot enough to burn you. Compare one side with the other for differences in each part. When it starts pulling hard, park it and look underneath.
This is a routine with me when we are towing cross country. When we stop, I get out and feel the tires and wheel centers. A low tire, hanging brake or dry wheel bearing will make heat before it fails, as a general rule. At the same time I can check hitches, chains, tie downs and lights.