Albion had been chasing the pair for weeks but had to wait to finalise the deals after lengthy negotiations with County's administrators.
Irish midfielder Dicker, 22, was on loan at the Withdean last season and manager Russell Slade told the club's official website he expects him to be a huge hit.
Slade said: "The fans will already know about Gary. We only suffered one defeat when he was here last season and he played a massive part in us staying up."
Centre-back Tunnicliffe made his Stockport debut at 16 and has attracted interest from Southampton and Liverpool.
"He covers the ground well for someone so tall," added Slade. "His feet are improving all the time, he has an enormous throw and he is the type of person and player that will be high on our agenda as we look to increase our squad."