You will be screwed if you suddenly get a job (any job) and need to start, and everything has a months long waiting list.
I think it comes down to whether you want to work or not. It sounds like you do. So I'd take the spot. You can always drop it in a few months of the job hunt isn't successful. You also probably have first dibs on a full time spot when you need one, if you take the PT spot.
ITA with this. Especially the "take the PT spot to be first in line for a FT spot" bit -- that's what I had to do with my DD when our first daycare fell through the week before I was supposed to go back to work. Worked great.
I think the bigger question is whether you (1) actually want to send your kid to daycare and work, (2) don't necessarily want to work and send your kid to daycare but are feeling the strain of your situation and think that getting of the house and working and getting some daycare will provide you a sanity break, or (3) don't really want to work or send your kid to daycare but feel like you should because you're afraid of future employability.
1 and 2 are perfectly legitimate reasons to take the daycare slot. 3 is your fear speaking, and is probably overstated; you are young, you have plenty of time in front of you to both work and raise your family.
And if this daycare is meh, you can always stay on the other lists and change over when another slot becomes available. None of this has to be permanent; it just has to be good enough (on all fronts) to get you through "now."