From the sound of it, the problem might be with the solenoid failing and maybe even the starter drive remaining engaged to the flywheel. That might explain the smell you described.
The solenoid is on the starter, on the bottom as installed. It's an electro-mechanical rig that the positive cable from the battery connects to, as well as the contacts for the ignition switch.
My suggestion to you would be to start with a rebuild, exchange unit or replacing the starter outright as a complete unit, solenoid and all. Almost the same work as changing just the solenoid and deals with the unit completely.
Other than jacking the car up high enough and lying on your back to get at it, this is a fairly straight forward job. And, again, youd have to do this to get at the solenoid.
The starter is left side bottom back of the engine, just about at the chassis line.
After removing the dust cover (if still there, it's a plastic can-shaped jacket over the starter, solenoid and positive cable from the battery), depending on year and model, there are the battery cable connection nut and 2 blade connectors to remove. Then the 2 bolts that secure the starter to the flywheel, top and bottom of the starter's flange.
The lower bolt is fairly easy to access, but a tip for the upper one is to have an assortment of extensions for your ratchet and kind of snake it in along the top of the starter so the wrench is in front of it, rather than trying to get at it from top, over the fender. I use a 3/8" drive air ratchet for this job since it requires no swing room. This has been a good investment overall and has saved my knuckles considerable pain.
This is also probably one if the few times that the oil filter facing upward can be appreciated, as it is out of the way of this job.
With the bolts and wires removed, the starter pulls back, actually sliding toward the front of the engine, and drops out once it clears the lip of the bellhouse.
Installation is the reverse. Stock starters can be found rebuilt/exchange or outright sale both through the catalogs or, in some cases, through local parts stores. The latter being easier if there's a core return involved, as well as being able to compare your old unit to the new one at the store.
Aftermarket high-torque starters are also available through the catalogs.
I hope this helps you. Let me know how you make out.
Safety Fast!
Art Isaacs