Hi Bryan,
There are a number of other things that could be the problem, or contributing to it. Besides the other excellent suggestions you've already got (which primarily point to the carbs and very correctly recommend you get hold of a good service manual), a good, old-fashioned mechanic might help you diagnose problems. Even if you do the work yourself, it's often helpful to have some help finding the problem when you are unfamiliar with this sort of work. Most shops are happy to do this for about a 1 hour labor charge. And, many mechanics actually appreciate people who want to learn about and work on their own cars. Now, a lot of young, modern-trained mechanics might not be able to find their way around an old-style engine compartment (where do I hook up the computer?) so I'd suggest you look for an "experienced" shop! They can run a few tests and really help narrow down the possibilities.
Yes, it certainly could be that the carbs are out of adjustment. (A good shop's exhaust gas analyzer can quickly help identify that.) But there are a variety of other possibilities that come to mind as well and can be checked out pretty easily.
For one, I'd want to run a compression and/or leak-down test on the engine to get some idea of the condition of things inside. You mention smoke in the engine compartment, and that might indicate blow-by coming out of the crankcase vent. Blow-by might be evidence of carbon build-up on piston rings or worn rings and cylinders. It could also be a vent hole plugged up or something simple like that. Or, it might be some oil getting past worn valve guides and into the cylinder, where it burns and produces the smoke, as well as carboning things up and fouling plugs. A leak-down test can help identify the problem, if there is one.
There's also a possibility the valves are receding gradually due to using unleaded gas, which is mostly a matter of re-adjustment until the recession gets pretty extreme. (Note: The TR 4-cylinder head is actually relatively resistant to this problem, more than many people give it credit for. It holds up pretty well with unleaded gas unless it's driven hard and fast. So, don't panic if it's never been converted to an "unleaded". Eventually it will need exhaust valve seats, different valve guides and stainless steel valves installed. But, that job can often be put off for a long, long time and many miles.)
There could also be an ignition system fault such as old wiring or slight mis-timing or worn points, rotor and cap on the dizzy. Or, it might be as simple as old spark plugs, excessive plug gap or the wrong type of spark plugs being used.
Back to the fuel supply, a less-than fully functional (i.e., worn) fuel pump can cause some backfiring or missing, similar to what you describe. Also, the needle valves and floats in the carbs need to be working properly and the carbs themselves need to be in synch.
Overall, it's a good idea to make sure everything else is working right *before* starting to adjust and fuss with the carbs. They are usually about the last thing on the tune-up list... after the ignition system, fuel supply, internal engine condition, valve adjustment, etc. are all checked out, corrected and adjusted as necessary.
Doug's question about the color of the smoke is an important one, too.
Finally, how long have you been driving the car? I ask because some roughness and valve noise is really pretty typical with these old engines, when cold at startup. Folks who haven't driven vintage cars like this, who's only real experience is with cars made since, say, the 80s, might find some of the typical noises, smells and smoke of older cars a little disconcerting, although it might not be a problem at all! The real proof is how the engine runs once it's fully warmed up.
Hope this helps!
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