Many states have a Bike/Pedestrian Program as part of their DOTs. You would need to go to that state's Department of Trans website and search for the B/P program. Then look to see if they have a designated bike route map online. Warning - many of these are little more than red, blue and green lines that have little connection to reality - but they are better than nothing. There are also traffic volume maps at state DOT sites - these are far better for knowing busy roads, but give no information on shoulder width.
Not online but Garmin's City Select allows you to do this. A bit expensive around $110 just for the software but it is meant for a GPS which will set you back another $400 or so. The City Select is really nice in that it lets you route right on the GPS.
There is an additional feature that allows you to avoid specific roads. I have set the routing to bicycle but it still picks roads I would rather avoid. So I can use this feature to avoid those roads.
You might be able to pick up a Metro Guide on Ebay for around $50 or so. It has similar capabilities to City Select but it does not allow you to route on the GPS, that must be done one the PC then loaded to the GPS.
If you plan to buy used on Ebay then be certain to ask how many unlocks remain on the software. Garmin gives you 2 per CD but if you don't plan to buy a GPS then you don't need to worry about the unlock (you don't need a GPS to use the software on your PC).
Having a GPS is very nice for just going out exploring and not having to worry about finding your way back. Of course you can do this with a map but the GPS makes it so much easier.