No reason why you can't use your Album In A Day songs in FAWM as well and taking on a small focussed 24 hour project might focus you to get a few more songs out than you otherwise would.
It's just a case of dropping your quality control entirely, and ignoring the internal voice urging you to quit after every song. I didn't get any songs out of it that I'm particularly proud of, but it was a fun experience. When I finished recording mine, I was in a really great mood so I went out to the pub to celebrate and met a really nice girl - and we're still together now, years later.
So there you have it - writing an album in a day is the only way to find true love. FACT.
My 4 songs on day 1 of this FAWM aren't going to pay the bills either...but the next day, I had a wicked-good collaboration session with a great friend. Not as great as your collab mind you...
For me, the trick is to not be too precious about my songs...especially the first few. I have to write whatever is there, without judgement.
I've only managed 4 songs in one day so far, a far cry from 14. So my limited personal experience is:
#1. Don't worry about song-smithing things to death, because
#2. Getting in the zone is easier when the internal critic is kept in it's place
#3. Real innovation is in there...might be deep down at times...but can be uncovered, so
#4. Just get out what comes out, and keep digging
The first few songs might be tripe, but embrace them as a gift, and count them towards the total