Mason Williams played it solo (albeit accompanied by band and orchestra) and there are plenty of tabs online.
I can't vouch for the accuracy of the Songsterr version, but maybe it sounds "thin" because you don't have the orchestra behind you?
You can see Williams playing it here, which should explain a lot:
https://www.youtube.com/watch?v=EEzyrpfrPEI
- he's not even playing at all in the bridge!
This looks like a good demo, worth comparing with online tabs:
https://www.youtube.com/watch?v=nHje3Fh4GJo