Revs not dropping when you pull in the clutch when warmed

First time I used a hair dryer to soften them up and some washing up liquid around the edges.

 

That sounds as if this could be your problem with the Carburettors over fuelling engine so much its getting into engine oil you certainly do not want petrol in engine oil you sort of loose the lubricating factor the engine needs to stop it locking upneedle and seats or a sinking float comes to mind, try removing plugs see what side is sooted up.

 

From what I've read, pods are notoriously difficult to set up.

 

mate you will be wasting your time and cash trying to get pods working. I have been round these T300s a while now and on various forums they try but fail. One trick I do know is you can cut the filter vertically down the centre so you have 2 halves,it makes the job much simpler but obviously it could potentially create a problem too. You can use silicone sealant on the join but not everyone bothers. I know the original part is difficult to fit but Triumph used all their knowledge to make it as efficient as it is,no one , believe me no one gets them running as well.
The fuel leaking into the sump is , as you have found out already, either a sticky float, worn out o rings ( I would do both while you are in there) and/or the fuel tap. It is a well known problem on our old girls.
You can get the parts from Sprint
http://www.triumphparts.gbr.cc/
or pay World Of Triumph exorbitant prices for original bits

 

sprint also do a rebuild kit for the petcock. too late i know

 

Sprint do list the bits for the thunderbird as well as the other T3 bikes, was looking in there only the other day. It dont matter anyway, to be fair you are doing it the easiest and best way,it just costs a tad more.lol

Fingers crossed for you mate.

 

Yeah - good luck with this Wayne.

 

Get that hair dryer out Wayne - you must own one with that long hair!