Glad you got the DR running again, sounds like an ethanol problem - its horrible stuff, I avoid using any fuel with it, in every petrol engine I own - if I have to use it and let the bike sit for any more than a few days then I use a fuel stabiliser (Sea Foam) which seems to help reduce crap building up in the carb.
Take that small white filter from the carb inlet and throw it in the bin, they have a tiny surface area and clog up quickly, instead use a proper inline fuel filter, they have 10x the surface area to filter the fuel and last a lot longer before having to clean them (backflush)
As for the decal popping on overrun, the DR is renowned for it, what setup do you have on the carb/airbox? its hard to tune it out with a performance can like the Barratt - or even the GSXR 1000 can, the only real way is to increase fuel flow, which will take the edge off it, but not much, if you do try that, increase the fuel in small stages and compare fuel consumption, but the fuel screw should not be out more ~2.5 turns.
Best way to clean the carb is to strip it down, remove all plastic and o rings and immerse it in Pine Sol for about 3 hours, this removes any varnish/gunk build-up.
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