I am in the same position, having to move due to our landlord selling (AGAIN!). We have only been here a year but prices have already risen since then. I am constantly checking rightmove and zoopla. Also if a property has been on the market a while you can ask if they would accept a lower rent amount.
We also have pets. Whilst they are part of our family and it would literally break my heart to rehome them. If it came down to a home that was just about in budget or waiting for something pet friendly I would choose my children. So yes try and find somewhere for all of you, but also research what your options are for rehoming so you are prepared for if it does have to happen. Eg I have spoken to my local animal shelter and I know they will be well looked after until they found another home.
Also go through all your direct debits, subscriptions and outgoings. Reduce anything you can now ready as a fall back for increased rent costs or towards removals. We've cancelled charity donation, Netflix, disney plus, TV license, Spotify, my contact lenses (back to cheap glasses!). I've reduced the Internet cost. No longer get the bus Home from work except Friday. Food, toiletries, household costs I've really cut back on. No more impulsive buys, go down to the cheapest brand and work up, more vegetarian meals, utilise olio for free food, yellow label for the freezer and batch cook for lunches.
We are decluttering everything, anything that hasn't been used in the last 6 months is being sold/donated. Technology, clothes, toys, homeware, kitchen devices. This will help as less to pack & move, and if you are put in emergency accommodation less stuff to find a storage solution for.
Today we have got together £280 worth of old computer games, consoles, broken phones, dvds ready to list for sale. 6 cookbooks I don't use and a bin liner of outgrown clothes from dd2 bedroom. Ive sorted the food cupboards so we eat everything there before buying more. I don't think I need anything except fresh for around a month.
I understand its really hard but if you can find 1 hr per day to sort things out it will be far less overwhelming when you come to packing ready to move. (Trust me this is move 5 in the last 11 years, the less you have the easier it is!)