The latter was abolished in 1819 in Act to abolish Appeals of Murder, Treason, Felony or Other Offenses, and Wager of Battle, or joining Issue and Trial by Battle, in Writs of Right, 59 Geo. 3, ch. 46 § 2 (1819)(Eng.).
The former [which I imagine to be the adversarial paradigm of American legal practise] was abolished in 1833 in An Act for the further Amendment of the Law and the better Advancement of Justice, 3 & 4 Will. 4, ch. 42 § 13 (1833)(Eng.).