Can you expedite looking for a property for yourself? If so then you could agree to childcare for a few hours a day Monday to Friday.
However, to the here and now. If possible, and it may not be, take a fixed timeout for yourself daily, perhaps 12 until 3. I presume but don’t know, that the weather is good. Could you take some sketching items and sit in a park, beside a river, in a botanical garden, museum cafe and do some drawing. As money is tight, take a small rucksack with a picnic. Can you bow out and go to bed or at least your room at 9 pm. This will give you quiet time and you family privacy. As adorable as he is make sure you alternate bath time, bedtime etc for your grandson. I hope you have a comfy chair in your room and your family knock before entering.
If you moved out to rent a one bedroom or studio flat until you buy, would your family receive some government help?
I really feel for you. I have had to say to family that I will not help financially unless there are literally NO jobs. They have had to compromise with retail or bar work instead of their particular choice. It also looked better on a CV that they were in employment, keen to work and be independent. It is about personal integrity. Employers prefer people who are flexible and will turn their hand to an alternative when needs must. After all what would your family do if you lived also where and could not help? Buying your own place and putting a time limit on the situation will force their hand. You also do not want to run down your capital. Time for a/nother calm conversation before there is a big falling out. Don't be around to be a referee. Two fit young people should be helping themselves. If your SIL is proud he should be compromising to support his family himself.
I wish you well. Come back and talk to us.