I would definitely dry the wood before covering it up, otherwise, MOLD in the ceiling. While it is open check with screwdriver for rot or decay and look for mold. Inside check for water that came through to the ceiling or wall and water damage. Smell for mold. Consider taking up the roof and replacing the fiberglass insulation and/or wood as needed. You could drill a hole and stick in a finger to see if the water got through and the ceiling insulation is wet or moldy.
If you are suspicious of inside moisture go for a dehumidifier and a fan. Don't expect much because much of the paneling and ceiling material is a plastic skin over a thin bit of plywood or pressed wood. It may not let the water out. In that small space a commercial dehumidifier would probably not make a difference.
On the whole roof replacement, why? Is the rubber roof bad or just the where it meets the edge or some bad roof job? If the roof is good, re-caulk and re-install the edging as needed.
Keep a tarp, ropes and bungees handy in case of unexpected rain. Watch out for dew overnight so you might as well tarp it at night in any case.
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