Swaim enrolled Butte College where he was converted to tight end and was a two-year starter, helping the team win back-to-back bowls and the 2012 Nor Cal Conference Championship. 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 22 starts out of 26 games, 13 receptions for 84 yards and one touchdown.
Swaim was selected by the Dallas Cowboys in the seventh round of the 2015 NFL Draft. The Cowboys traded their sixth round pick in the 2016 NFL Draft to the San Francisco 49ers in exchange for a seventh round pick after it became apparent Swaim was considering signing with the San Diego Chargers if he had become an undrafted free agent. Swaim was the 17th tight end drafted in 2015. He played well enough in the preseason to force the team to make the unusual of move of keeping 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. In 2016, he became the backup and blocking tight end after James Hanna missed the season with a knee bone bruise. He started the season opener against the New York Giants as the second tight end. He started in the ninth game against the Pittsburgh Steelers as the second tight end, but suffered a pectoral injury and was replaced by Escobar. On November 16, he was placed on the injured reserve list with a pectoral tear. He started 6 out of 9 games, making 6 receptions for 69 yards. In 2017, he missed part of the offseason because of a foot fracture he suffered during a personal workout. He was the third-string tight end behind Hanna. He was declared inactive against the Atlanta Falcons because of a knee injury he suffered in practice. He appeared in 15 games, posting 2 receptions for 25 yards. In 2018, Swaim entered the season as the Cowboys starting tight end after the sudden retirements of Jason Witten and Hanna. In the third game against the Seattle Seahawks, he had a career-high 5 receptions. In the fourth game against the Detroit Lions, Swaim scored his first professional touchdown on a one-yard reception from quarterbackDak Prescott. In the fifth game against the Houston Texans, he had a career-high 55 receiving yards. He started nine games before suffering a broken bone in his wrist in the Week 11 contest against the Atlanta Falcons. He missed the next four games before being placed on injured reserve on December 22. He was used mostly for blocking purposes, finishing the season with a career-high 26 receptions for 242 yards and one touchdown. He was a solid blocking tight end during his career with the Cowboys, that had his development limited by injuries.
On March 15, 2019, Swaim was signed by the Jacksonville Jaguars. He began the season as the backup behind James O'Shaughnessy. On October 22, he was placed on injured reserve with an ankle injury and a concussion. He appeared in just 6 games with 2 starts, posting 13 receptions for 65 yards. On March 17, 2020, Swaim was released by the Jaguars.
