"s'adonner au proselytisme de cette methodologie" isn't actually a sentence, and a complete sentence is always best.
Lack of a complete sentence makes this difficult, but I would probably just treat s'adonner au proselytisme as amounting to "proselytise". Just as effectuer le ... is usually best translated just by using the English verb for whatever noun it is in English.
Otherwise, possibly "engage in proselytizing". I don't think s'adonner à commonly has this "devoted to" meaning at all; it just means engage in or be involved in.