It looks like the conflict between the FAQ and Kevin's word of mouth rulings is still in effect. Namely, the FAQ entry of

Q: If a Location card tells an investigator that he may move to
another location and have an encounter there, what happens if
there is a monster and/or gate at that location?
A: Assuming that the investigator chooses to move and have
an encounter, he must then first successfully evade or fight any
monsters at the destination location. If he is successful in doing
so (or if there was no monster), he then carries out what amounts
to a normal Arkham Encounters Phase at the new location,
following either the “Gate” or “No Gate” instructions on pages
8–9 of the rulebook.

This version of the FAQ has been around since, well, forever. Before Kingsport, at least. In any case, Kevin has said that this was supposed to be changed. From the Official Answers from Kevin Wilson thread:

Note (06/24/08 ):
I have received an answer from Kevin, and i copy here his answer verbatim:

So his previous ruling, which limits combat with monsters on the board only during the movement phase was the good one. Too bad the Kingsport FAQ reintroduced the error, but at least we have an official ruling.

KevinW wrote:Hmm, looking at the files, now that I have a chance. It appears that the ruling was changed after FAQ 1.2 and the FAQ stuff that went into Kingsport was simply not updated in that instance. Stick with the combat only occurs during the movement phase ruling. My apologies for the confusion.

-Kevin Wilson
Fantasy Flight Games

So, once again, we're in need of an official response. Is the game correct, or is the game designer?

You noticed that too, huh? They must have created the Innsmouth rules by modifying the Kingsport book

"Stick with X" is the only ruling I need to see. That's the way I've been playing it, so until a newer ruling contradicts it, I'll continue to play that way.

I am sure that Kevin will confirm his ruling (that you don't fight monsters on the board during the Encounter phase). Too bad this was overlooked in the Innsmouth rulebook. It is time to make a big general FAQ/Errata now!

Yeah, whoops. Stick with the updated ruling, not the cut n' paste error in the Innsmouth rules. We're gonna be working on a new version of the FAQ soon though, so don't worry.