I had not finished my post, but had to do stuff in RL. As the parent of a SN child, imo, its partly up to the parents to make a social life and out of school activities happen for DC.
Is DS in a special school? If so, have you gone round the parents of the other children in his class at parents' evening to create a contact list, with their permission, so you can organise class picnics and activities in the school holidays; and Mums' Nights Out - I did it for DD's class through primary school? There was an unspoken rule among the parents of DD's class of ten, that we all invited the whole class to DC's birthday party whether our child liked all of them or not, because they weren't likely to get any other party invitations, and it was the only respite most of us tended to get. Or, set up a support group for parents of children in the school.
I set up a support group for parents of children with DD's primary SEN in my county.
If he is in mainstream, have you joined the Parent Carer Forum - there should be one in every LA? Ask there or look on the LA's local offer to see what SN groups and activities there are? For instance, a parent near me set up an inclusive special needs drama group for SN children (starting with her son) and their "normal" siblings. Another set up SN trampolining sessions on a Sunday morning at a local sports centre - well that might not be suitable for OP's DS; but its up to her to think of something he and other children could do?
As for OP is not likely to get respite, because DS's needs aren't severe enough on paper - what paper is this? I suggest if OP has not done so already, she ask for an assessment of need for DS under S17 of the Children's Act; and a carer's assessment for herself. When she has got the drafts of those, go on the special needs section for further advice on what to do next. For instance, my DD was offered as a teenager, a buddy service whereby a "normal" young person would take her out to activities for their age, like ten pin bowling, the cinema, etc.