You had said in an earlier post that you had replaced the starter relay. This is a term used for the Starter Solenoid Switch it's located just behind the battery and bolted to the inner fender. Did you replace this Item? If yes, then you have already replaced the solenoid. I am assuming you have the most common engine for a Thunderbird in 68, a 390? The 390, 428, and 427 FE engines will have a Solenoid like I have just described, if you have a 429 the solenoid will be on the starter like a GM. If this is the case, Buy a Starter w/solenoid already attached, it will save you a headache.
Either way you should do it yourself. IT IS THE ONLY WAY TO LEARN! You should not have to take the exhaust down unless someone has made a change in the exhaust routing. If you do have the 429, make sure that you disconnect the batter first.