To be honest I don't know anything about your particular sector so please treat my bletherings with caution - for that reason I'll speak generically and hope that something from this is useful.
The NUJ (National Union of Journalists) has some general advice on how to handle late payments, take a look at Advice - General / Late and problem payments.
Artsadmin Anchor is a fortnightly newsletter for artists and art professionals. It's an interesting mixed bag of job ads for front of house theatre people and artist residencies as well as calls for writing (both paid and unpaid). I've not seen people offering their services but they might do that if you ask them (I'm not involved in the art world but sometimes there's an art / science crossover that I want to tell science chums about).
Sign up to Sian Meades-Williams' Freelance Writing Jobs - it's free but paid subscribers get access the day before.
In the science / medical world there's an umbrella list of major charities - the Association of Medical Research Charities. Perhaps there's something similar in your sector? Wikipedia has a page of Arts organisations based in the United Kingdom that might start you off (but will probably be rather time consuming to search!) - I'm imagining you might contact organisations, or haunt their vacancies pages.
Museum Jobs might not be as helpful but possibly worth skimming - they tend to be job-jobs rather than freelance-contract-y things but I can see one is looking for a digital content producer, so right ballpark I suppose.
The UK Gov's Contracts Finder service contains all sorts of contracts over £10k but is a bit fiddly to search - to be honest I don't really understand much of what's there but you might make more sense of it, it's not really my area. It has absolutely everything (I once came across a council's request for some trees and hedging).
In the short term universities are unlikely to be much use to you because their payment systems are legendarily slow. They also generally have in-house writers. They are recipients of art-based funding, particularly from the AHRC (Arts & Humanities Research Council) and they do write about it. Universities also employ freelancers to teach their staff to write so that might be a possibility. They also employ communications trainers although I'm only familiar with science communication (I assume there's an 'artscomm' world parallel to my 'scicomm' world!).
It's not just universities that get arts funding from AHRC (though to be fair it mostly is, but also V&A and music colleges etc) so possibly some projects would employ a freelance writer in some capacity - have a look at https://gtr.ukri.org (this tells you about all projects funded by UK Research Councils but doesn't include charity funding etc). Type in AHRC then select currently open projects and tick the box for AHRC-funded projects too to filter. It just gives you an idea of who's being funded to do what. Projects typically want to 'woohoo' their project to the world and not all will have in-house woohooers.
Jisc hosts the Jiscmail listserv, a bunch of academic mailing lists for various communities of practice. I run one for people who do science communication, so it won't be much use to you, but there may well be other lists that would be more fruitful. Have a rummage in some of these categories, many of the posts will be public so you can get a sense of what they talk about -
Art & Design: https://www.jiscmail.ac.uk/mailinglists/category/index.html
Performing arts: https://www.jiscmail.ac.uk/mailinglists/category/W4.html
Humanities: https://www.jiscmail.ac.uk/mailinglists/category/Humanities.html
Note the W4 in the Performing arts link above - you can replace it with any of these alphanumeric codes to look at mailing lists in that sub-topic https://www.jiscmail.ac.uk/groups/startagroup/categories.html
Jo