If you have bike that's fitted for that kind of terrain should be possible to cycle Trail of the Eagle's Nests. That trail is mainly unpaved roads with a lot of sand, so bike suited for that terrain is recommended (wide tyres, etc).
Also there is a bike version of that trail - it is quite similar to the main one, but in theory more suited for cycling. Here you can find Wikipedia article about this one, and the route is here.
While I said it's possibly more suited for cycling, I don't really have good comparison between them. I cycled a lot in the general area, so I've been on some parts of both of them, but that was quite a while ago. On both you can probably find harder (more sandy) parts.
This trails are maintained (in terms of painting markers and selecting the route by PTTK (Polish Tourist and Sightseeing Society). They are usually pretty well marked in the terrain, but some markings might be missing so have a map. Also the fact the the trail is for bikes does not mean that it will lead you on the paved roads. Quite often you will ride through forest paths and roads, so something better than city bike or road bike is recommended.
Lesser Poland has a good map of marked trails here provided by PTTK, which can be found here (click on "Mapa", and then you can change language to English). There is also accompanying Android app (I think also available in English). In options you can adjust it to view only bike trails.
Unfortunately there is no such thing for Silesian region - the best I was able to find is this, but it's not very detailed and not sure how up to date it might be.
You can also buy paper map like this one, which claims to have the bike version of the Trail of the Eagle's Nests marked (and also other bike trails in the area).
For finding places to stay you can take a look at my answer here. I recently used one of the portals I mention there - nocowanie.pl, for the first time and had pretty good experience. For accommodation I would recommend finding it beforehand, as looking the same day might be problematic especially in some less touristy places.
I found some sources mentioning that Wislana Trasa Rowerowa I mentioned in comments is mainly asphalt/gravel, and it's not too hard. If you google the name, photos seems to confirm that.
Also you don't really have to follow marked trail - there is plenty of calm roads going through small villages with barely any traffic. Just take a look at the route you want to take (for example to more or less follow Trail of the Eagle's Nests), and then check the local roads on Google Maps. Poland has really good coverage in StreetView photos, so you can check what type of road you have where and see how busy it is.