Welshman attended Siena College of the Metro Atlantic Athletic Conference his freshman and sophomore years in college. He was named the MAAC rookie of the year in 2009 after recording 6 goals and 3 assists. His stellar sophomore season saw him jump onto the national stage after recording 13 goals and 3 assists in only 16 games played. He was awarded MAAC offensive player of the year award, all MAAC first team honors and all Region first team honors. He transferred to Oregon State University of the Pac-12 Conference for his final two years of college. He was named to the All-Pac-12 first team after his junior year in 2011 after recording 3 goals and 7 assists. He was again named to the All-Pac-12 first team his senior year after recording 10 goals and 4 assists. He was one of 54 seniors invited to the 2013 MLS Combine.

Welshman was drafted sixteenth overall in the 2013 MLS SuperDraft by Toronto FC. He made his debut for Toronto as a second half sub in a 1–0 away defeat to Vancouver Whitecaps FC. He started and played the full 90 in Toronto’s 2–0 home leg victory against Montreal Impact in the 2013 Canadian Championship. After failing to make another senior team appearance Welshman was released by Toronto in February 2014.

He spent 2014 playing for Sigma FC in League1 Ontario.

In February 2015, Welshman joined the preseason camp of Real Salt Lake as a trialist. He ultimately signed with their reserve team, Real Monarchs.

Welshman is eligible for Guyana through his parents. In January 2015 it was reported that he was invited by Guyana coach Jamaal Shabazz to an upcoming training camp ahead of a match against Barbados. He made his debut on February 1 against Barbados and scored a goal.

In January 2016, Welshman was promoted and signed a deal with Real Salt Lake. His contract option was declined at the end of the 2016 MLS season.

On January 3, 2018, Welshman signed a contract with USL side FC Cincinnati. On May 16, 2018, Welshman scored his first professional hat-trick in a 4–1 victory over Detroit City FC in the second round of the 2018 U.S. Open Cup. He re-signed with the club ahead of their entry into MLS for the 2019 season on December 10, 2018.

Cincinnati loaned Welshman to Forge FC of the Canadian Premier League for the 2019 CPL season on March 8, 2019. He was recalled from his loan on August 2, and his contract with Cincinnati was terminated by mutual consent.

In August 2019, Welshman signed with Hapoel Haifa of the Israeli Premier League.

In Guyana’s final match of the CONCACAF Nations League qualifying tournament, Welshman scored the match winner against Belize, sending the nation to the CONCACAF Gold Cup for the first time in their history. In May 2019, he was named to Guyana’s provisional squad for the Gold Cup. He was named to the final squad on May 30.