Will Time Machine transfer my problems?

Hi,

I have a question about using Time Machine to transfer my old computer contents to a new computer without also transferring the problems the old computer is having.

I've been using a Macbook Pro (OSX 10.6.8, 2.4Ghz core 2 duo, 4GB ram, 256 GB) for about 5 years. I just bought a new Mac Air (OSX 10.9, 1.3 Ghz core i5, 8GB ram, 256 GB). The old computer runs pretty clunky with lots of beachball-of-death (I'd filled 220 of 256 GB space, so I just transferred 75GB to an external hardrive and cut it down to 175 GB).

My question: I want to use Time Machine to backup the old Mac Pro and transfer it with all programs and files to the new Mac Air. Do I also risk transferring all the gunk (background applications and stuff) that's making the computer run slow? Is there a way around that?

Thanks