I have recently done almost exactly the same thing. In my case my domain was registered to, and by, another person (a friend, not Plusnet) but Plusnet hosted both my email and my website. I hope the following helps.
The first step is to take ownership of the domain. This is managed by a Domain Registrar. If you are staying with the same registrar it SHOULD be easy to change ownership but, in my case, although myself and the person managing my domain did everything that should have been necessary it just would not transfer but changing registrars from IONOS to 123reg proved to be very easy and quick. IONOS also had tried to help. Now I have full ownership I could easily change registrars again.
So far there was no lost of service while the domain registration was changed! I still pointed to the Plusnet name servers.
The next stage was to move my website to a new host. In my case this was not the same organisation as the Registrar. The data was easy to do and changing the DNS records to point to my new website was then easy. I took the opportunity to add the necessary certificates to enable HTTPS support which was also easy. I never tried to do this with Plusnet. I then needed to change the DNS records so that the web services pointed to my new website but the email temporarily remained with Plusnet.
The final stage was to create new email boxes on my new host and then change the DNS records to point everything to the new server. That was also easy. My email and web site are now hosted on an entirely different server but also now under my full control.
The only hard part was finding how to remove my domain email from the plusnet server. I asked for help via this forum to do that. There appeared to be no other way. Actually the request had to be made via a private message to a Plusnet admin who i had corresponded with beforehand via this forum and he was kind enough to do that for me. It probably would have done no harm to leave them there however.
I now ONLY use Plusnet for supplying telephone and basic broadband but I can now move both of these services without disrupting my web and email or standard telephone. With the recent DNS issues I am looking at whether to change ISP and ditch Plusnet completely but unfortunately I still have a year to go on my phone and broadband contract. I tried to determine what the penalty would be (with the recent poor service being the reason for the change) but have not yet had an answer. They offered a discount but, with my current referrals discounts, that amounted to just £0.40 per month saving!
I have already ditched Plusnet for my mobile phone and have a contract at far below the price of the discounted rate on EE that Plusnet were offering and now pay just £3.90 per month on a rolling monthly contract.
Plusnet used to be so easy to deal with if any support issues arose! I had no issues recommending them to anyone who asked. Regrettably it is evident that this is no longer the case. When I initially joined Plusnet in 2003 they were an excellent service provider!
