"Courtin' in the Rain" was sung by T. Texas Tyler. His recording is available on "Best of T. Texas Tyler," Collectors' Choice Music CD #51, 1998; and on the anthology "Louisiana Hayride, Vol. 2," Music Mill CD #70016, 2000. Authorship is credited to ["Dock"?] Walsh.

It is listed in the Folk Music Index as "Courting in the Rain." The recordings listed are:
Carolina Tar Heels on "Carolina Tar Heels," Folk Legacy FSA-024, LP (1965), cut# 3
Wade Mainer on "From the Maple on the Hill," Old Homestead OHTRS 4000, LP (1976), cut#B.03
Wade & Julia Mainer, on "Best of Seedtime on the Cumberland," June Appal JA 0059C, Cas (198?), cut# 9
"T" Texas Tyler on "Country and Western Album," Forum 7G 501, LP (196?), cut# 5

honkingduck.com gives this information:
Recording: Columbia 15075-D
Date Issued: June 1926
Side: B
Title: We Courted In the Rain
Artist: "Dock" Walsh
Recording Date: April 17, 1926

Couldn't find the lyrics, though.