@AwaAnBileYerHeid, your distinction between the "deserving" and "undeserving" poor is offensive and divisive. A lot of people who have never worked have no chance of ever finding work.
OP, if the council have placed you in temporary accommodation that is as rank as yours sounds, then get on their case about sorting the damp and vermin and don't let up. The temp accommodation officer should be your starting point, and if you get nowhere with them, the environmental health department is your next port of call. (FYI, I once worked for a council where an EHO threatened to issue disrepair proceedings against their own council - that got them sorting out the problem pdq!) If the council drag their feet, then get your local councillor and MP involved, start a formal complaint and that doesn't resolve things, complain to the ombudsman.
Remember that in situations like this, those who shout loudest get best results.
I really feel for you, I used to work with homeless families and some of the properties they have been placed in have been utterly vile.
Are you bidding on properties? If you've been placed in TA because you were homeless, or about to be, you will be very high priority for housing. That may well cut down your waiting time. A lot of councils are building council homes and housing associations are also getting more funding for new homes than they have for a long time. When they're built, lots of families will get rehoused all at once. When that happens, some homes go to those with relatively low priority.
I'm not surprised you're feeling shit about your situation, but this is won't last forever. When your DS gets free nursery hours, that will help, and when he starts school, that will help even more. You'll be able to work more hours and with your degree, you're better placed than most to get a better paying job and progress in a career.
This too will pass, and things will improve. Just try and see it as a bad patch and you will come out the other side.