Swaim attended Pleasant Valley High School, where he played linebacker and received All-section and All-league honors as a senior, after registering 124 tackles (65 solo), 6 sacks, one interception, one fumble recovery and 2 blocked extra points. He also practiced basketball.
After his sophomore season, he transferred to the University of Texas where he was a two-year starter, being used mostly as a blocking tight end and on special teams. He finished with 26 games (22 starts), 13 receptions for 84 yards and one touchdown.
Swaim was selected by the Dallas Cowboys in the seventh round (246th overall) of the 2015 NFL Draft, when the team surprisingly traded a 2016 sixth-round draft choice (#178-D.J. White) to the San Francisco 49ers to assure obtaining his rights, after he was considering signing with the San Diego Chargers as an undrafted free agent. He played well enough in the preseason to force the team to keep four tight ends.
As a rookie, he was declared inactive for 12 games, while playing in 4 contests. He had one start against the New England Patriots, when he replaced an injured James Hanna and the Cowboys opened in a three-tight end set. The season-ending injury to Gavin Escobar, gave him a chance to play in the last 2 games and record one reception for no yards.
Swaim was placed on injured reserve on November 16, 2016 with a pectoral tear.