two birds in hand beating bush:

1) woodshedding is first attested in the punishment sense. but to woodshed was coined in the musical sense!

1936 L. Armstrong Swing that Music 71 We used to practice together, ‘wood-shed’ as we say (from the old-time way of going out into the wood-shed to practice a new song). 1946 Mezzrow & Wolfe Really the Blues viii. 108 I'll have to woodshed this thing awhile so I can get straight with you all. 1950 Blesh & Janis They all played Ragtime (1958) x. 203, I would hear the tunes and, to make sure, go home and ‘woodshed’ them in every key, put them in major and minor and all the ninth chords. 1968 A. Young in A. Chapman New Black Voices (1972) Drew's got an alto [horn].+ Drew dont hardly touch it, he too busy woodsheddin his drums. 1978 Amer. Speech 1975 L. 302 [Jargon of barber-shop singing.] Woodshed, work out the harmony parts (to a known melody) by ear; sing as a group for the first time+; improvise (an interpretation).

Hence "woodshedding" vbl. n., (a) the dispensing of punishment; (b) the practice or rehearsal of music; (c) spontaneous or improvised barber-shop singing.
1940 Amer. Speech XV. 205 Woodshedding, disciplinary action. 1946 Mezzrow & Wolfe Really the Blues ix. 151 Instead of woodshedding, he went out after the big money with the primitive equipment he had when he started. 1955 Shapiro & Hentoff Hear me talkin' to Ya xi. 190 It was here that the term ‘woodshedding’ originated. When one of the gang wanted to rehearse his part, he would go off into the woods and practice. 1956 S. Longstreet Real Jazz xiii. 101 Bix [Beiderbecke] did plenty of woodshedding, playing alone, to some recording on the family Victrola. 1973 T. Pynchon Gravity's Rainbow i. 129 No head falsetto here but complete, out of the honest breast, a baritone voice brought over years of woodshedding up to this range. 1974 Harmonizer Jan.-Feb. 18/2 Woodshedding is not a ‘spectator sport’—only participants can fully enjoy it. 1976 Times 27 Sept. 12/4 Spontaneous barbershopping is known as woodshedding, because a woodshed is as good a place as any to burst into sudden song.

2. as I said, well nigh a year ago (or was it more), the baseball origin of southpaw certainly seems plausible. but there is that nasty citation* from 1848 to deal with.

1. A person's left hand. (In quot. 1848, a punch or blow with the left hand.) 1848 Democratic B-hoy, ‘I say, Lewy, give him a sockdologer!’ ‘Curse the Old Hoss, what a south-paw he has given me!’ 1885 Sporting Life 14 Jan. 4/3 They had always been accustomed to having their opponents hug their bases pretty close, out of respect for Morris' quick throw over to first with that south-paw of his. 1942 Berrey & Van den Bark Amer. Thes. Slang §121/53 Southpaw, wrong hand or fist, the left hand or fist. 1948 Chicago Tribune 20 Apr. i. 20/5 He waved his big south~paw and ducked under the roof.

2. One who pitches or throws with the left hand; a left-handed person.
In Boxing, a southpaw leads with his right hand.
1891 Chicago Herald 24 July 6/1 The new south-paw+came to town yesterday. 1911 Daily Colonist (Victoria, B.C.) 15 Apr. 8/5 Davis came up to bat.+ He faced the twirler right-handed. He always does with southpaws. 1932 Ring Apr. 5/2 McCoy was a slow southpaw who had proved just a good workout for Joe Chip. 1942 Berrey & Van den Bark Amer. Thes. Slang §430/10 Left-handed person,+south-paw. 1947 J. Gunther Inside U.S.A. xl. 657 Ah won't even go to the Polo Grounds unless a southpaw's pitchin'. 1951 Sport 6–12 Apr. 8/2 On the same bill, Joe Lucy, the young southpaw, meets South African lightweight Gerald Dreyer. 1955 Sci. News Let. 14 May 310/2 The family cat may have a preferred paw+, and pussy is most often a southpaw when she is not ambidextrous. 1959 Sunday Times 8 Nov. 32/6 In the ball parks all over the United States the so-called ‘diamond’, formed by the track between the bases, is always oriented to the same points of the compass, so that in whatever park a team is playing the pitcher on his mound will always have his right hand on the north side of his body; hence a left-hander is a ‘southpaw’. 1967 Boston Sunday Herald 26 Mar. ii. 7/1 Rocket Rod Laver leads the greatest tennis show on earth into Boston Garden Monday night.+ The freckle-faced southpaw is the top-seeded player. 1970 H. McLeave Question of Negligence (1973) vi. 48 ‘Nobody told me he was a southpaw.’ Even the psychiatrist had+forgotten that the surgeon cut with his left hand. 1976 Billings (Montana) Gaz. 26 June 1-b/2 The 6–0 lefthander, the only southpaw listed on the Angels' roster, struck out six and walked the same number. 1976 ‘A. Burgess’ Beard's Roman Women (1977) v. 110 Donatella, a south~paw, animated this [sc. her left shoulder-blade] while lifting the one remaining chair from the front room. 1978 M. Kenyon Deep Pocket ix. 103 He wore shorts and boxing gloves. ‘'E's a southpaw,’ Peckover said.

3. attrib. or as adj. Left-handed; also transf., left-footed, and fig.
1891 Cricket 29 Oct. 463/1 The Germantown man returned the ball like a flash to the wicket, and the ‘south-paw’ batsman was run out. 1932 J. T. Farrell Young Lonigan iii. 126 It was swell for Studs to play,+knowing he had made that good kick,+to run back and pick one of Helen's southpaw kicks out of the air. 1949 Sun (Baltimore) 3 June 18/8 They would have been bunched against southpaw pitching. 1957 R. Watson-Watt Three Steps to Victory xliii. 245 This was, however, a south-paw kind of compliment. 1969 New Scientist 6 Nov. 277/2 Jack Bodell has just become the first south~paw heavyweight champion in British boxing history.


*http://loc.harpweek.com/LCPoliticalCartoons/IndexDisplayCartoonMedium.asp?SourceIndex-
=People&IndexText=Cass,%20Lewis&UniqueID=80&Year=1848