Well Elissa, you have certainly set yourself a challenge.  Yes, flour is not a good idea and using clay on top is likely to exacerbate the mould problem.  Methylated Cellulose is really the best gel glue to use, mixed with about 30% PVA but next best is wallpaper paste, also with PVA added.  Don't dilute it any more than thick cream consistency and use it very sparingly.
There are many ways to approach this but in terms of time and drying, I suggest using as much dry material as possible to get your basic shaping.  You can scrunch newspaper into the shapes you want and bind it with masking tape.  Cooking foil also works very well but has a cost. 
Having built up your shape, then use pasted newspaper to bind it all.  Damp single sheet newspaper is very weak so one way to improve this is to lay a sheet of newspaper on a flat surface, lightly paste it and place another sheet on top at right angles (warp and weft), then another and another.  Tear pieces from this mat to lay on your sculpture and allow to dry (hairdryer helps).
Are you sure about fur?  A possibility would be to cut feather shapes from thick paper and lay them over each other.
If you use a method like this it shouldn't matter too much that you have already used flour in a paste, as long as it is fully dry before you apply the subsequent dry material.
Good luck and please let us know how you get on.
David Osborne