Hi Hamish,
Yuck! What a mess. Welcome to the world of ridiculously complex GIS data sources!
My first idea would be regarding the TIGER data; perhaps that was in UTM with a NAD 27 or NAD 83 datum, and you got the wrong one when tdoing transformations? Is the change consistent over the map, or is it 10 feet in one place 200 feet in another? If it is sort of consistent then I would guess that you have a datum problem (we have about a 50 metre variation between the two in New Mexico).
Another option is to rubbersheet the two sets of data; here you assume one is correct, and then pick points on the other map; you might pick 36, and then the software stretches , compresses and twists the second set of data to match the first. Let me know if you want more details.
If there is any way that you can get the attributes from the TIGER data into your street centerlines, then that will be the way forward. Having such data attached as attributes makes your maps 10 times more useful. If there is anyway you can manipulate the attributes to have a common field, say by changing the column names, that will be the best way.
Sorry for such vagueness. Feel free to ask for more details.
Cheers,
David Walsh