Rehydrating clayey soils in the lab can be a bit tricky. I would not dry the soils completely before running moisture-density relationship tests.
As for determining field moisture content, I would do a correlation between oven-dried and Speedy, then adjust accordingly when using the Speedy in the field. You might also consider checking a dessication moisture content (drying to constant weight in a dessicator, rather than an oven). Oven drying, particularly at required standard temperatures, can be a bit misleading in some clayey materials.
As for the field, you might consider some lime treatment of the soil. If you do so, make sure the lab tests reflect this as well.