Green’s Dictionary of Slang

possum v.

[SE possum + possum n. (1), i.e. the habits of the animal]

to dissemble; to feign sickness; thus possum trick n., a feigning of injury to lure an intended robbery victim.

[UK]T. Flint Geography of the Mississippi Valley n.p.: As one who counterfeits, or dissembles strongly for a particular purpose is said to be possuming [F&H].
R. Levinge Echoes from the Backwoods II 32: ’Possuming is become an idiom; a term signifying any one who is humbugging or deceiving.
[US]M. Griffith Autobiog. of a Female Slave 140: She ain’t crazy, only ’possuming so as to shuffle outen the work.
[US]Knickerbocker (N.Y.) lvii (June) 627: This last looked like affectation, or, as the negroes call it, possuming.
[US]Bartlett Dict. Americanisms (4th edn) 485: Possum. To feign, dissemble. An expression alluding to the habit of the opossum, which throws itself on its back, and feigns death on the approach of an enemy.
[US]Chicago Inter-Ocean 6 Feb. n.p.: A possibility of possuming among those [i.e. grizzlies] stretched out below.
[US]S. Ford Shorty McCabe 35: The Lamb only lasted half-way through the seventh, and ’possumed the count at that.
[US] ‘Jargon of the Und.’ in DN V 459: Possum trick, Highway robbery as follows. A man lies on the side of the highway pretending to be injured. A kind-hearted motorist stops, puts him in the back seat and goes for a doctor. The man, seeing his opportunity, strikes his benefactor over the head with a blackjack.
[UK]R. Frede Entry E (1961) 155: Gordy was back—lying stretched out asleep or dead or possum near the door.

In phrases

possum-foot (v.)

to tread quietly.

[UK]L. Hadow Full Cycle 153: The men were drifting off to the camps, [...] or possum-footing across to the women’s dormitory.