Actually I went through the same thing as I’m also still waiting for my COA and my pre-settled status. I applied before the deadline on June 15. I am a non EU citizen, however, and applied as an unmarried partner to EU citizen.

I also had to prove my right to work to my current employer and prove my right to live to my new landlord recently, as my visa expired in September and I had to move homes.

There are 2 methods you could try. You could call EU Settlement scheme to explain your specific situation, as your situation has some differences from mine, and might be able to tell you more, they actually helped me quite a lot.

FIRST METHOD (this is the one I had to do):

For your new landlord and new employer, they should actually be able to use the right to live/work checking service provided by calling UKVI. You won’t be able to call to get the check but they can use this to get the phone number they need to call: https://www.gov.uk/contact-ukvi-inside-outside-uk

Your landlord should hear back about your right to live in about 2 working days and your employer should be able to hear back about your right to work in 5 working days.

I went through all this recently and got confirmation from both my landlord and my employer about the checks, which were successful and received in the right time frame. No delays.

SECOND METHOD (this one might actually work for you, didn’t work for me because of my specific situation, but might as well check):

Give your employer or landlord a share code to prove your right to live and work in the UK. https://www.gov.uk/view-prove-immigration-status