Need help transfering system software from old macbook to newer one.

I recently purchased a used Black MacBook running OS X 10.5.8 and came with the 10.5.2 install disks.
The computer that it is replacing is running 10.6.8 but I don't have the disks anymore.

I want to transfer the entire system, along with all of the applications to the new computer.

I believe that these MacBooks are otherwise identical.

After using Migration Assistant to move the Time Machine backup, my apps are there, but most of the disk images won't mount and it still claims to be running 10.5.8.

You can not transfer a newer system to an older one. It won't work. You need to either find your Snow Leopard disk or order a new one from Apple. Apple is still taking orders by phone to their customer service line. It's $29.99 plus local tax and shipping.