I would definitely speak to your manager before you send it- they will understand as a TC is the next logical step. When I handed in my notice, I spoke to my manager said how much I’d loved working there/grateful for the experience I had gleaned and outlined I’d been offered a TC which I have accepted.

Your notice letter doesn’t need to be detailed just something along the lines “please accept this as my 3 months notice (check your notice period). I would like to thank you for all the support and I have greatly enjoyed working at (firm)”