It won't be sent through until you're processed as an actual leaver, which will be July.
Start looking for alternative roles. No-one will be at all concerned that you're on garden leave, it happens a fair bit and it often happens once senior people or those in critical roles hand on their notice. So a new place wouldn't be worried.
Make sure the agreement they are sending through has a sample agreed reference attached for your review and approval. Also check if they will cover the cost of a lawyer's review of the document too.
I was definitely not 'senior'! although that was in my job title. we got taken over by a much bigger firm a year ago and they had to intergrate me into a team by law (TUPE?). And it was a very very uneasy fit. To the point where when I threatened legal action they initially offered to pay me my two months and put me on gardening leave but i told them to fuck off, told them I wanted 6 months so they added the extra month ex-gratia and we agreed to call it quits that day. for all our mental health's sake. it was pretty hellish.
2 job interviews lined up though and ive been open about me being on garden leave to them. i just wish i had a fucking garden!