The earliest printed source I could find is:

Quit dat ticklin' me: June 7, 1873
Dime Song Book, No. 32: 388

From a collection of songs published by Beadle and Adams in 118 of the 121 Songsters described in The House of Beadle and Adams," Vol. I, pages 380 to 395

Her's some info from folk index:

Quit that Ticklin' Me [Me II-Y 13]
Rt - Trouble in Mind
At - Ain't That Trouble In Mind
1. Blue Sky Boys. In Concert, 1964, Rounder 0236, LP (1989), trk# 2
2. Carson, Fiddlin' John. Fiddlin' John Carson. Vol 4, Document DOCD 8017, CD (1997), trk# 23 [1927/10/11]
3. Volo Bogtrotters. Tough Luck, Marimac 9042, Cas (1991), trk# 20
4. Volo Bogtrotters. Old Time Music on the Air, Vol. 1, Rounder 0331, Cas (1994), trk# 9

Quit that Ticklin' Me (Tune)
Rt - Sally Ann/Anne
1. Cole, Calvin. Old Originals, Vol. 2, Rounder 0058, LP (1978), trk# 15
2. Kimble Family. Carroll County Pioneers, Marimac 9036, Cas (1992), trk# 22 [1973-77]