Britton was a key player in their 5-0 thumping of Bradford City at Wembley.
He told The Sun: "It is still hard to believe what Swansea have achieved. It is now only just sinking in. But not everything was straightforward.
"But I was too hydrated and my sample was clear. It took me until 4.30pm to give a sample that was acceptable.
"After the final and that amazing result at Wembley, I was asked to give another urine sample. I couldn't believe I was asked again. I took my own magnum of champagne in there. I also took two bottles of beer.
"I ended up a little bit drunk. By the time I got to the changing rooms everyone had gone. I was gutted. I got in there hoping a few people would be around but there was just the kit lady and a physio. I missed holding the cup and all the good pictures."