"Purposefully" has a strong ring of intent or determination about it. "He was hell-bent on finishing painting the room before dark so he set about the task purposefully." I can't remember ever hearing anyone say "purposely" this end and I think it could be a bit of an eyebrow-raiser, though* its antiquity can't be denied. I think we tend to say "on purpose" or "deliberately." I've seen "sewerage" on my annual bills from the water company but I haven't heard anyone saying it (in which case it would almost certainly be being used in error). Sewage is always in the news here as a scandalous item, and we have Surfers Against Sewage, who are very high-profile, so that's the word we're all conditioned to use this end.

*I could have said "albeit" there, I s'pose... ;-)