Well done for taking control of the situation.
There are a lot of people on MN who are just never going to understand anyone's life or difficulties that are too different from their own.
Looking forward for the future, it may be worth investing in a cheap junior hacksaw and a Stanley knife to chop stuff up with.
It's more time and energy to expend, but a lot of surprising things can be made to fit domestic bins and recycling bags, or turned into sizes that comply with public recycling bin rules etc. (I'm lucky enough to have graduated to a cheap reciprocal saw, it's life changing!)
Most council 'X item' collections accept things like say dismantled kitchen unit pieces, and a dismantled bathroom cabinet, all taped together neatly, as being one item. General rule of thumb is, can you pick it up and carry it as one lump, and book it as 'assorted wood/ laminate' Xcms x Ycms. Sadly for the environment, cheap brown parcel tape works best on packaging them as a 'one item.
I know it's often not possible, but IME approach dustmen discretely holding a fiver or a tenner, and disposal problems can vanish as swiftly as the money.
As pp said, skip bags are always a good shout. Sometimes local waste companies do their own cheaper versions.
Another one that might or might not be useful for the future, but a bit more complicated:
if you have any friends/reli's who can drive, and depending where you live: look into zip vans and similar.
You join up for £19.50 for a year, which then gives you access to a van for £9 per hour or £84 per day (weekdays) or £10 per hour or £95 per day (weekends) and 60 miles inc fuel.
It can get complicated booking in to some tips with rules about vans, so check, but I found if you book under a known reg no, and then change to that vans reg details once known on the day, it works out. Most need you as the resident to be a passenger with your ID, but once you've figured the systems out, it works well for people with ongoing needs, and is worth the initial headache figuring out how to make the systems come together.