Does an offer have to be in writing at the first instance or can I
confirm in writing on my return? How will the estate agent of the
target property know that I am genuine?

Anything written or spoken other than exchange of contracts is meaningless.
Exchange will happen simultaneously with every property in the chain when
everyone is happy to do so. (Actually not in my case, as I like to keep
chains bumping along, so exchanged with my purchaser before my vendor). You
simply state the position of your buyer and the state of proceedings to your
vendor. If they find that acceptable along with your offer price, then you
can start to arrange conveyancing.

Christian.