Get on to your MP and your local councillor. They will be able to kick ass (especially the MP, councils tend to shit their collective pants when MPs get involved.
Keep it simple, bullet points with the dates (or approximate dates, where actual dates aren't known) shown and the sequence of events in date order. Copy in whoever is the most senior bod at the council involved in your complaint.
It would also be worth checking if you are "statutory overcrowded". Shelter has a very good guide to working this out england.shelter.org.uk/housing_advice/repairs/check_if_your_home_is_overcrowded_by_law . If you are (although I think it unlikely, as u-10s don't count, iirc) that puts you in a much stronger position. "Statutory overcrowding" may have been what the housing officer was referring too when they said you weren't overcrowded, but the council's bedroom allocation/property size criteria will be different from statutory overcrowding. They need a bollocking for misinforming you, if nothing else.
This is shocking maladministration imo. Even if your older daughter isn't counted as living there, you are still entitled to a 3-bed property (both boys in one bedroom, daughter in the other, or 2 youngest sharing and oldest in his own room). If your 21-year old counts as living there, then you're entitled to 4 bedrooms.
You would be entitled to an extra bedroom if you can persuade the council that your younger boy cannot share a room because of his various issues. You would need good medical and social services evidence and, ideally, something from his school as well. If he has an HCP, that may contain useful supporting evidence.
BUT - even if you got the best possible outcome, that would only put you at the front of the queue for the next 3 (or 4, if they decide you need 4) bedroom property that becomes vacant. There may be homeless families in B&B that come before you in terms of the council's allocations policy. Their policy should be publicly available, if it's not online, email them and request a copy. Work out what your banding should be according to their policy.
Finally, make sure you include somewhere that your housing situation the lack of room is endangering your children and severely impacting on your ability to keep them safe. This is another thing that can galvanise them into action for fear of ending up in deep shit if something dreadful happens and there is a serious case review.
I'm so sorry you're going through this, it must be awful. This sort of situation, and the feeling of hopelessness that goes with it, is why I changed jobs a few years ago, but I learned a few tricks along the way.
Good luck, OP.