This one is hard to prove. Unless it’s actually written somewhere - or their’s clear history that the longest-servers always get first refusal for the full time posts, it’s not certain your MS was the reason, as nobody’s automatically entitled to any job.
Realistically, I agree it looks likely it was the MS, but they might just claim you were unlucky, and that the other candidates were better for some reason, even though they were newer (did they have more experience elsewhere, for example?)
If it were me, I think I’d try to keep the two issues separate, otherwise you risk confusing things. Tackle one at a time - first the immediate one of the “reasonable adjustments” for disabled people employers are required to make under the Equality Act 2010 (this supersedes the Disability Discrimination Act, which covered a lot of the same ground). Importantly, the Act expressly identifies people with MS as covered by the disability legislation from the moment of diagnosis - i.e. there is no test for how disabled they (that is, we!) have to be before they are covered - it’s automatic from the start.
So your employer couldn’t argue you are exempt, because you’re “not really very disabled”, for example. It makes no difference whether your disability is obvious in everyday life or not - the fact of being diagnosed is enough.
So you need to get to the bottom of WHY being permitted to sit - or even lean, as you were - would not be a “reasonable adjustment” you are entitled to. I can’t think of any good reason. It wouldn’t cost much, even if they were to buy you a chair, and if there are no safety issues, such as those I suggested earlier, like blocking a fire exit or something, I really can’t see what grounds they’d have for refusing. “It gives a wrong impression” certainly wouldn’t be a good enough reason. If you can’t stand for as long as fit and well colleagues, then it’s not showing preferential treatment or setting a bad example for you to be allowed to sit. Their only other objection would be if it was provable you wouldn’t be able to do the job adequately in a sitting position, or that it was dangerous in some way. I can’t think of any reason either of those would be true, but then I’ve never worked in a laundry, so I don’t know exactly what’s involved (from what you’ve said, it sounds like you’re ironing) and whether there are any risks or limitations of doing it sitting down.
I do think it’s a good idea to get your occupational therapist involved, so personally, I would not give final agreement to anything on Monday - even if it sounds good - like saying you can have a chair. If (as I hope you will) you get to the point they offer that, say how pleased you are, but that you would like to involve your occupational therapist to make sure the type of chair is suitable, and see if there’s anything else she recommends. IF they would actually be buying the chair, you can dress this up as helping them, by saying you’d hate them to spend money on the wrong thing, so you’d like to run it past her to avoid a mistake.
So unfortunately there’s probably going to be a bit of hedging for time going on on Monday, because you want to consult the experts before agreeing to anything - so don’t get pressed into a snap decision.
I think the full time working is a separate issue, and once (hopefully) you’ve got the reasonable adjustments sorted out, you can argue that now there are arrangements in place to overcome any difficulties, you are even keener to be considered for a full time role. There’s no way they should be saying you can’t because you’re ill, but I think that’s a fight for another day, and one thing at once. Get it OKed you can sit down first.