Welcome. A few helpful pointers;
1. You would most likely get more responses if you post in the Sport and Touring>T3 Sport / Touring forum. Your bike is a T3.
2. There is a search feature up near the top of the page. Type in a few key words related to your inquiry and hit search. You might find what you're looking for. Once you get familar with the search funtion you can refine it with an advanced search. Some examples of what you might find:
http://www.triumphrat.net/daytona-de...-96-d1200.html
http://www.triumphrat.net/t3-sport-t...ride-home.html
http://www.triumphrat.net/t3-sport-t...l-related.html
There are also "stickys" at the top of most forums and the T3 has a FAQ sticky.
3. I didn't find what I was looking for when I just searched but if you search my name and all my posts you might find something I wrote (along with others) to newcomers about basic sequential troubleshooting dependent on your issue.
For you I'd check fuel delivery first.
Check to see if you have enough gas.
Try using "Prime" position on your petcock (vs On or RES). The petcock is vacuum operated but when on "Prime" feeds the carbs via gravity (lets you know if maybe the petcock is bad). I just reread your post; If your petcock is already on "Prime" you may be flooding the carbs.
Check all your fuel and vacuum lines to ensure they are not cracked, kinked, missing, etc. Bottom line; eliminate fuel issues as the source of your troubles then move to the next step (Which should be ignition but I don't have a manual in front of me.)
4. T3 known issues (not all related to your problem and not meant to scare you off.) These issues are covered extensivley in the T3 forum and I'm sure I'm missing a few:
Vacuum petcock
misrouted fuel and vacuum lines
bad coils
sprag clutch
alternator bolt
Good luck with your quest because these era Triumph bikes ROCK!
I forgot to add; since I own a 95 Daytona 1200 I frequently search the Daytona Deliberations, T3, AND Trophy forums