You have a 3 year old, they will be in school soon enough which will reduce childcare fees, could you then start putting some of that away?
You cant go full time now but could you look at asking your boss or looking out for full time roles?
I think its more looking ahead and seeing a way out in future that might help.
I think also getting signed off work in the short term, so take the sick time to get the doctors appointment to get signed off.
Depending on where you are (I know you dont want to say) you can find rentals with zero deposit but you have to look for them.
I think the options are currently to either say something and plan together which you dont want, or make future plans and move slower. Even making small progress might help you mentally feel better.
Things like childcare arent forever and that will free up funds, pay can increase, your hours may be able to increase or you may find a new job, but if you can start moving towards any of those things it may help you feel this is possible in the future.