Since HLSW Gamers Search server lists don't work any more, the work-around is to use the Steam server list as source. Copy the servers you want into your Steam favs, and then import them into HLSW. This is done through the server list menu in HLSW, Import, Steam favs. It doesn't automatically find the correct path so you have to manually open this file:
C:\Program Files (x86)\Steam\userdata\***numeric-account-id***\7\remote\serverbrowser_hist.vdf
Replace ***numeric-account-id*** with one of the subfolders of "userdata". Each refers to a different Steam account.
Storing the server list probably won't work well, better tell it to make a temporary server list instead and then copy all servers from the new temporary server list into a normal server list.
Next time you do this, the last used path is remembered, and you can click on the arrow / triangle inside the file name field to select the last used path + file name.