Welcome to the forum and congrats on picking up your T-Dub.
The '87 used a seperate lighting coil and the lights would only work if the bike was running. Based on your comments it sounds like your problems may be electrical however there could also be some underlying mechanical issues as well which can complicate solving your no start issues. Sit tight and I am sure someone more familiar with the 87's can suggest a few things to check before you bring it to a mechanic. I hope you get it all sorted out soon so you can again enjoy the new ride!