Only just spotted this so the places I'd recommend have already been mentioned but I thought I'd share our perspective with you since we go up as a band and, like yours, we have a lot of arranged stuff.
Having been to the festival for the last eleven years our typical day has become:
Morning - recover (see evening, below), take the kids to the festival play group thing and get along to whatever workshops seem interesting. There are eight of us when we're all there but the wide range of workshops on the programme meens that there is, pretty much, always something for each of us.
Afternoon - assuming there isn't a concert we fancy, more workshops or go along to one or other of the programmed sessions. We're a varied bunch - some of us are fully fledged tune session types who would generally go to the sessions in the Ship or the Board, others are more takey-turny singer types who would tend to go to the Fleece where there are often 4 sessions on at once, some more raucous and anarchic than others. There's usually, even this year, something outside the Middle Earth if the weather's ok. There are also some programmed slow sessions - in the middle Earth this year - which those of us who don't have a head full of tunes go to fairly often.
Evening - depending on what time we get out and, again, assuming there's no concert we like the sound of, we'll often go somewhere to listen to others for a bit, the Endeavour being a favourite, and then maybe go and join a session or start one somewhere. As Nick said, above, the Golden Lion is often good at this stage. There is usually a polite session, mostly songs with guitars & so on but some tunes, in the front room with people taking turns and quietly listening to each other which is nice but not really our bag though we do go in for an hour or so on occasion. The back room is generally full of non-musicians but they tend to welcome and encourage people to play if they see instruments and this regularly develops into a boisterous song & tune session.
Later evening - inevitably, somewhere between 9 ish and 10 ish we repair to the back room of the Station. This has become our nightly destination for the past 7 or 8 years. It's generally very noisy and very crowded, sometimes meaning a seat doesn't become available for quite a while but, if that's the case, it means there'll be music to listen to!
The nature of the Station - large-ish room, lots of non-folky types in there drinking, the fact a lot of the programmed acts often go down there after fulfilling their duties, the 2 pm last orders and the fact it's pretty rowdy - leads to a varied experience with no two nights in there being the same, unlike a few of the other places. Depending on who's there it will sometimes be mostly tunes, other times mostly songs, other times anywhere between the two. People are generally happy to join in and be joined in with and there is pretty much always a substantial "audience" of non-musicians who have come in to hear the music. There are at least two groups of people I'm aware of and have become friends with who go to Whitby that week each year largely because of what's on in the Station.
So, if you want to play your stuff to for some people to hear and enjoy and for others to join in with, the Station's where it's at. Don't be expecting pin-drop silence, though. The people around the musicians will generally quieten down when a gentle tune is played or an unnacompanied song is sung but there will still be noise from closer to the bar and the other rooms so those precious about people hearing each subtle nuance of their 18 verse unnacompanied anti-foxhunting song will probably enjoy themselves more elsewhere, possibly the front room on the right which has unnacompanied singing all week long.