I live near Durham - The Brit is in Darlington which is south of Durham about 15 miles.
The Copper Beech is also in Darlington and is on a Thurs - not sure what is on at the time u visit but they have a web site "Darlington folk Club"
The Candlelite is on Tues, in Newton Aycliffe
And again not sure about week of your visit but usually an open mic nite. Its about 10 miles from Durham at the Cricket club. There is an "open" facebook page - The Candlelite DL5.
Durham Folk Club is on Thurs at Tap & Spile in Framwellgate Moor about 2 miles from central Durham. Its a Singaround club, like the Brit.