:D That`s a shame i liked "Jive Turkeys" !  :P 
A lot of cover bands record demos. There are some venues who insist on demos in order to get gigs.
I would suggest that you don`t record full songs but record small snippets say intro, verse, bridge & chorus. Then you can record  7 - 8  songs as it were & this will help give Landlords & agents a much better idea of what you sound like, without  taking up huge amounts of there time. Which there`ll apriciate.
 :D  8)