I think the singing/reciting plays & poems/reading the lesson in Church etc are somewhat different to the accent aspect. In all those cases it is somebody else's words, not yours, so your pyschological relationship with the words is very different.

Almost all the techniques for overcoming a stammer rely on your modifying the way you speak words. Virtually all of them work, but you (that is I!) have to concentrate so hard on how you are saying things that it is extremely difficult to say anything.

One common approach is what is called 'syllable timed speech'. This is simply to make sure every syllable in every word takes the same length of time, making you sound something like a Dalek. I recommend you all to try speaking that way for at least half and hour to discover why it both does and does not work. (It is, on the other hand, handy to use on occasions when things get too bad to cope with.)