I know the ISA rules will change next April, but in the mean time, if I transfer a cash ISA to another provider, and I haven’t yet used up my full allowance, can I then pay into the new ISA once the transfer is completed?
My logic says yes, but the govt rules say “only pay into one cash ISA per tax year” and I don’t know how (in)flexibly they interpret that.
Yes, but you have to wait for the transfer to complete.
Ok google
Yes - done it loads of times.
Thanks all 
To fully answer the original question, yes; you can only have one of each ISA type each year. So, you can have one Stocks and shares and one Cash ISA.
As noted, you can transfer them from provider to provider which is still just one of each ISA per provider.
The allowance is the same in that if you had £4000 in transfer left this would be the same.
You can open as many as you like but can only contribute to one of each type each year. This will change in April 2024 and you will be allowed to contribute to more than one of the same type subject to a maximum of £20,000.