[The general sense may be to cause to sway about, pull about, pull, and relation to SWAG is probable; but it is not clear that all the senses below belong to the same word.]
1. trans. To castrate (a ram) by tying the scrotum tightly with a string.
1663. Boyle, Usef. Exp. Nat. Philos., II. V. xii. 234. A Servant of mine that deals much in Cattle, and had lately divers Sheep swiggd (as they call it) after this manner.
a. 1722. Lisle, Husb. (1757), 315. Swigging, which is girding them hard round the cods, and cutting the cod away close to the string.
2. ? To pull about.
1684. Creech, Virg. Ecl., iii. The Lambkins swigs the Teat, But find no moisture.
1697. Dryden, Æneid, IX. 73. The bleating Lambs Securely swig the Dug, beneath the Dams.
3. Naut. To pull at the bight of a rope which is fast at one end to a fixed object and at the other to a movable one; to pull (a sail, etc.) up in this manner.
1794. Rigging & Seamanship, I. 176. Swigging off, pulling upon the middle of a tight rope that is made fast at both ends.
1827. Examiner, 154/1. Taking about a calendar month to swig up her mainsail.
1882. Nares, Seamanship (ed. 6), 57. Swinging or swigging off, that is, pulling at right angles to a taut rope.
4. intr. To sway about, waver; to move with a swaying motion.
1833. M. Scott, Tom Cringle, xv. Her long slender wands of masts which used to swig about.
1896. Kipling, Seven Seas, Rhyme Three Sealers, 8. The landward breeze Bring up the harbour noise, And ebb of Yokohama Bay Swigs chattering through the buoys.