Former Republic of Ireland midfielder Lawrence signed a two-year deal with Shrews after his release from Barnsley, where he worked with Mellon.
Mellon told the club's website: "I'm committed to building an exciting team for our fans and for the chairman.
"He has been really brilliant in helping to bring Liam in as it was an important signing for me and once again he has backed me.
"Everyone should know about the backing he has given me and his commitment to get this team together."
Mellon believes the signing of Lawrence, who has won 15 caps for the Republic of Ireland, shows the club's ambition.
The Shrews boss added: "Liam is a lad I know very well from my time at Barnsley and from during his career.
"He is a really good guy and a very talented footballer and definitely the type of player I wanted in the club to do that organising job for me on the pitch.
"Liam is a really exciting signing and one that shows people that Shrewsbury Town want to have a real good go at making us a good side."