Need to find a house!

[DaffodilsAndDandelions]

I have made the decision to leave my partner and take our toddler and I'm 15 weeks pregnant. I posted a couple of days ago that this has been on the cards for a long time but the process has been sped up.
Unfortunately I can't find anywhere in my area to rent. I've moved a lot since I left my mums 11 years ago and have found a lovely area, nice friends and know what I want from a property. The only problem is Covid seems to have decimated the rental market and there is nothing suitable available. I need a minimum of 2 beds, garden (preferably enclosed but I can always put up a temporary gate to take down when we leave), pet friendly (dog). I would love a fire/wood burner in a cottage type place. I would rather not live in a town. We are rural atm in a little village. I don't like new builds.
I can't even find anything meeting the necessary criteria, nevermind the wants list. Also must not be more than £600pcm.
Any advice??

Bump

What area would you like to live in. What are the priorities like shops and transport. Are you working or claiming benefits that can help towards the costs.

I am north north yorks. I will be travelling north to Teesside uni but all my friends live slightly south. I drive so don't need to be on public transport routes. I don't like shopping so that's not important either. Tesco currently deliver it for me. I don't need to be near a school as I Hope to buy somewhere in 3 or 4 years before toddler starts schools.

There is really nothing suitable on rightmove at the moment. How long do I wait before just moving anywhere even if not suitable? Before Covid there was 40+ properties to choose from. Now there is 10.

I need to claim UC but can't until I start uni, move house. Can't move house until I get my student loans and bursaries confirmed as income.