"Yes" the park roads might still be wet so avoid the marsh roads and select sand ridge tracks as easier when wet.
Parks must be booked in advance and camping fees paid. Note that some camps have been privatized and are no longer reserved with DWNP.
A good guide is
http://www.maunselfdrive4x4.com for reservation info.
Once booked with any operator you still have to pay park entry fees with DWNP at one of the pay points.
And "yes" you can carry meat from Letlhakane to Lekhubu and Nata and up to Kasane as well as back to Maun when traveling through the parks.
Not so if you use tar roads from Kasane to Nata, and to Maun.
Note; There is no road from
Nata south to Maun but I think you referred to Kasane - Maun via Moremi.