With large or small firms it can indeed take a long time to get the offer in writing as then it is legally binding. Until you have something in writing you have nothing.
You should not be turning down other opportunities. Your optimal strategy is to continue to pursue all options and say nothing to any of them about any other options you are pursuing. If any of the other parities close an opportunity then reconsider your options. Until then you should try to maximise your upside which requires you wait until the job offer in writing. You should hold the university place open for as long as possible even going beyond their admissions deadline neither signing nor pulling out and still saying you intend to take the place.
The reality is that the university won't go out of business if you delay past their deadline and then take the job else withdraw. They are competing for talent with other universities and employers and they know it. Their worst outcome is no where near as bad for them as a bad outcome is to you. That's a guess on my part based on some knowledge and experiences; but the fact it is an estimation is salient. You do know the cost of a bad outcome to you; you don't know the cost of a bad outcome to them. They might not even be able to accurately figure out all their options as they may have multiple second choice candidates any of which may pull out at any point else fulfill the place months after the deadline.
Putting up a deadline and having you pull yourself out of the program at the deadline (possibly to end up with nothing) is a convenience to them. Don't do that. It is for them to figure out whether they pull the place from you at the deadline and offer it to a second choice candidate else hold the place open for you a while longer whilst you do whatever it is you are doing before you sign on the dotted line. It's their call. They are trying to push the ball to you to optimise their outcome. Their deadline is a strategy that they have setup to deal with the situations that arise such as the one you are in.
It is not for your to try to optimise the outcomes of other parties you are negotiating with. They negotiate for them; you negotiate for you. They look after themselves, have lawyers, professional admissions staff, a policy document, experience of such things. You have you looking out for yourself. Do so. Good luck.
(p.s. Sorry to point this out, but you might be the second choice candidate for your dream job, and they are waiting for their first choice candidate to accept or reject. Perhaps their first choice candidate is out of country and is having visa application issues and that could take many weeks or months to resolve. Perhaps that other candidate will not be able to take job and the job will come to you. Second guessing all the situations and outcomes doesn't change the optimal strategy; hold out for the dream job. Your best outcome is still your best outcome. Your worst outcome, that both opportunities fall through, don't actually mean much over the course of your career. You got short listed for your dream job and a doctoral place at a university. You will be get to where you want to be. The one thing you don't want at the end of life is regrets. So aim high.)