We had this exact situation around 4 years ago. Undertook a mutual exchange from a tiny house that we had fully decorated carpeted and importantly kept clean, to a much needed larger house.
The house we exchanged too was soaked in aninal urine and a fair amount of (hidden behind furniture) faeces.
We eventually figured out the black dots covering the walls were fly poo! New neighbours still give me a strange look after watching me mop the walls with bleach 🤣
On moving day it transpired they hadn't actually bothered hiring a van or arranging any help so we not only moved us in but moved them out. Most rooms were still full with furniture despite the fact we'd been in communication all along and they'd assured us they were moving things downstairs and packing up etc for ease.
They tried to leave a loft and shed full of stuff to collect at a later date! We helpfully delivered it all to them the next day.
You have to accept the house in the condition it is in and we were also advised we could not request any repairs for 2 years unless emergency such as gas and electrical. There was a broken window, a smashed sink and a leak in the kitchen that we had to repair ourselves.
We thankfully arranged for the children (toddlers) to stay out overnight as it would have been unsafe for them to be there.
We piled everything we owned into 1 room. Literally like it was still in a van. We lived from a suitcase packed with clothes and bare essentials.
Removed all carpets (hired a skip in advance, but they'd also left us a full one!) Anything that didnt fit in the skip was put in the front garden.
I hadn't realised just how bad the house smelt on viewing it (all windows open and liberal air freshener I expect) I nipped to tesco and blew an extortionate amount on plug ins and automatic air fresheners for every room just to make it bearable.
We slept on a mattress in the room we'd managed to strip back the best. We woke up covered in flea bites! We went out and bought flea bombs set one off in each room and left the door shut whilst we dealt with the other rooms.
The bathroom was black with mould. I scrubbed it with a mould spray and probably a toxic amount of chemicals tbh.
We took all blinds and any metal fixtures out of the house, they were all rusty it really was disgusting.
We had thankfully both booked 2 weeks off work and managed to do 16 hour days just stripping everything out and giving the whole place a whitewash. We'd already saved up for (cheap) flooring throughout the upstairs and paid slightly more for someone to fit a decent laminate throughout the downstairs.
Looking back at photos now I'm not sure how we did it all! We both cried on the first night wondering what on earth we had done, but 4 years on and pretty much every room has been redecorated and its bloody lovely. It was worth the work and I'm really hoping to buy it in the next few years.
As well as the suggestions listed by PP make sure you have a number of screwdrivers and tools to hand.
I hope your swappers at least arrange their own van!
Good luck!