How can we give this gift to our daughter properly?

My husband and I want to buy a used car for my daughter for her new job. This car costs$ 20,000. I know gift limit for 2022 is $16000/person. So my husband and I together can give up to $32000 without fill any form when we file the tax.

The checking account which we will use to pay the car is a joint checking account between my husband and I.

By writing out this check to buy the car, I will use up my gift limit while part of my husband's gift limit.

 

Do we need to write a gift letter to my daughter and keep record or it does not matter because it is under gift limit for 2 persons( my husband and I).

 

And I know we no need to fill any form when we file married jointly tax return in 2022, am I correct?