どうも here is like どうやら, and it pairs up with らしい to show that it's not entirely clear, but seems that way.
方 (かた) gets appended to the pre-masu form of verbs, so you'll never see it with の in front (as a suffix meaning "way of...", that is. It could mean "person", but that's entirely different). So 仕方 is used with 挨拶 because 挨拶 is a する動詞 (サ変動詞), and it's the する that conjugates to add 方, i.e., し方 (I'm pretty sure I've read that 仕 here is an 当て字, used for its sound rather than meaning, but the dictionary isn't backing me up right now). Then 喋る does its normal pre-masu conjugation and 方 gets stuck on there: 喋り方.