Vol. 8 of this version of Davis' papers at page 123 references a Mar. 30, 1862 letter from Van Dorn to Davis while en route to Corinth. It notes he was aboard the Acacia. This would likely indicate he was on the White or the Mississippi at the time.

http://books.google.com/books?id=TM_JRu2YXrgC&pg=PA123&lpg=PR36&vq=APril&dq=april+1862+van+dorn+jefferson+davis+letter&output=html_text