Are Offer Acceptance letters legally binding documents?

tldr - had an interview in february for a job which has been very flaky.

I signed an offer acceptance letter which specified that there would be a contract after they do their due diligence.
There was no communication for 8 weeks, no background checks etc) - they get in contact with me asking when Im starting.
I told them I havent handed in my notice as there is no contract yet (been screwed over before) and that notice period was 3 months.
they are demanding I resign asap as the acceptance letter is "legally binding" - even though there are no terms of employment, no start date, my referees havent been contacted at all by the employer, nor have they requested dbs checks etc. Ive emailed my professional union but they are useless (as usual). Ive also flagged it to the hr department of my current place who feel it is suspicious but will discuss it further tomorrow.

Personally I dont mind staying where I am at the moment (the reason I applied for a job in february was because there was a lot of uncertainty around my contract which was up at the end of march) and trust my luck, it was renewed a few days after accepting the offer for these lot. I honestly have a lot of misgivings about the plac[post="97010742"]cancel[/post]e I applied to but at the same time, I am aware my current contract extension may be my last (ends in october), thus am aware I need to look for work elsewhere. I want to decline the offer but want to do it in a way which keeps doors open as I wish to use 1 of the interview panel to help with a postgrad work placement for a course I have been trying to apply for the last 5 years, only to not be legible because I dont have a placement lined up.