Most likely what is burning is a resistor due to over current. In general I suggest a trip to the local stereo repair house that specializes in audio repair. Here in Dublin CA we have the Stereo Doctor. They are good and do all of my repairs when I don't have the time or inclination to do it myself.
Be advised, if it is a tube that is causing something inside the amp to burn, then swapping tubes will cause the other channel to burn! I suggest you simply purchase some new tubes and give that a try. If you don't need them now you will need them eventually, so the money is not wasted.
If it really was a resistor starting to smoke, then a short burn (seconds) may not have hurt anything other than cosmetics. Extended burning, for say several minute(s), may permanently damage the part.
Another possibility for burning is some previous owner put some damping rings or compound on the tubes to squish microphonics and that is what you are smelling. If that is the case, then the odor is no big deal. I would just remove the rings or compound and clean the tubes. As mentioned above, the dust that collects on tubes may smell a bit as it burns off. This is also a no big deal.
The fact you can't adjust the bias properly does indicate a problem with either the tube or the associated circuitry, though. Most likely that is the source of what you are smelling and it is the first place I would look for problems. If something was really damaged from over heat, then you should be able to see it.