Dry all studs and sheathing to 12% moisture (some contractor’s use 15%).

Here is the link:https://www.epa.gov/mold/printable-version-brief-g...It's a good read, whether or not there is the potential for lead based paint to be disturbed.Another booklet about "Mold, Moisture and Your Home" is also an essential communication that every landlord should be giving to every tenant.

If the existing floor is vinyl, the answer should be yes, you can because vinyl provides a good moisture seal. 

With the sun beating down on roofs year round, or leaves falling and sitting there causing moisture to go under the roof and rot wood, the usefulness of shingle roofs is not a lot over 10 years.  

Also, consider whether moisture is really an issue though.

The meter will detect the presence of moisture.

- (See Photo 1) - At the back corner of the cabinet below the kitchen countertop there is moisture damaged wood and drywall, there is some biological growth present on the drywall and wood here

The following are some property selection guidelines.Older properties require more maintenance than newer properties.Composition roofs require more maintenance than tile roofs.Properties in climates with hard freezes require more maintenance than properties in mild climates.Properties in locations with a lot of moisture require more maintenance than properties in dryer climates.Wood siding requires more maintenance than aluminum or stucco siding.Properties with lush vegetation require more maintenance than properties with little or no vegetation.Locations with high levels of termite activity will require more maintenance than areas with little or no termite activity.The tenant pool has a significant impact on your ongoing maintenance cost.

A bigger issue to consider is where the moisture is coming from and how to reduce it.

The joists are dry-rotted, could be all of them due to no moisture barrier between the dirt floor and subfloor.