As contracted captain Zach Merrett tries to exit the AFL club, the Bombers have endured weeks of turbulence following another barren season.
Their substantial supporter base is restless and ex-players are unhappy.
Essendon great James Hird hailed Welsh's appointment, saying it can unify the fractured club.
Instead of facing a potential challenge, Barham has quit the presidency and plans to leave the board entirely.
Welsh, who played 162 games for Essendon and has become highly successful in his business life, steps up from being vice-president.
"I have given my all over the past three-and-a-half years and tackled every challenge with everything that I have," Barham wrote in a letter to Essendon members.
"I took my responsibility to fight for and defend our club, our players, coaches and administrators extremely seriously.
"I'm proud of what we have achieved in transforming the club and making the hard decisions to set it up for long-term, sustained success.
"I have no doubt we are now on the right track, and I leave excited about the future.
"Andrew will be an outstanding president and it has been a privilege to work with him."
Within a couple of hours of Barham's announcement, Hird said Welsh can be who the club needs.
Hird and Welsh were teammates in the 2000s. While Hird is one of Essendon's greatest figures, he has not had a role at the club since he resigned as coach in 2015.
"With Andrew Welsh coming on board, it will bring the club together. He's the sort of personality that will get the club on the right page, everyone together and the support behind him," Hird said.
"If that's put together properly, on the field it will come together really quickly. It's a great footy club, it just needs to get back to performing well on the field."
Welsh released a statement on Tuesday afternoon, thanking Barham and saying "we are in a strong position.
"The work of recent years has given us stability and, most importantly, the right people in key roles.
"It's time to write our next chapter and if I have one message for you (club members), it's this: be optimistic, be excited and be part of this journey with us."
The president's departure comes after Merrett publicly addressed his future at the Brownlow Medal on Monday night, almost two weeks after he met with Hawthorn coach Sam Mitchell.
Merrett said he felt "shame" as he plans to leave the Bombers for a long-time rival.
"The Bombers fans are the best in the competition," he told the Seven Network.
"There's still things I need to work on to see where I end up.
"It's a bit of a shame at the moment, I am not there to upset fans, I appreciate them dearly."
Merrett is contracted at Essendon until the end of 2027, but his position at the club appears untenable even though the Bombers are publicly saying he will not be traded.
Contracted defender Jordan Ridley also wants out with his desire to reunite with departing free agent Sam Draper at grand-final powerhouse Brisbane next year.
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