Welcome to the forum!
Congrats (maybe?) on your TW. So sorry to hear about the problems that you are having getting it running.
Have you ever seen/heard this engine run? 1987's are known to have CDI problems that may present as carb problems. I am neither a carb expert nor a CDI/electrical expert, but I can offer you a few things. I have a known good '87 CDI and several new replacement carbs that others have successfully used on TW's, that you would be welcome to try out on your bike.
Since it sounds like you are local, maybe we should try to get another "Wrench & Ride" together and see if we can get your bike going?
Is yours a red one or a blue?
Let me know if you are interested,