Clapham said his position had become "untenable" following criticism in the media and on social media.
The decision followed on from the news that three directors, including Shrouder, would step aside in June.
Director Nigel Allen will take on the chairmanship on an interim basis while new investors are sought.
Clapham added: "(Directors) Nigel Allen and Liam Watson are considering their positions, Nigel has agreed to act as Interim Chairman until the 30 June 2017 during which time the board will seek a meeting with the Trust in Yellow Board to establish their views for the future as the call for change has come from them following the Open Meeting.
"The board will urgently seek to establish any level of interest in bringing in investors who may wish to take a controlling interest in the ownership of the club."
After their relegation from the National League was confirmed following defeat by Dover Athletic, Southport beat promotion-chasing Forest Green Rovers - only their third win of 2017.

Only 50.5 overs were possible at the MCG, but Australia still got wickets at regular intervals with Jackson Bird taking two for 53.
Azhar's 110-ball half-century saw him pass 1,000 Test runs in 2016.
Australia lead the three-Test series 1-0, having beaten Pakistan by 39 runs in the opener at the Gabba.
Sami Aslam was undone by Nathan Lyon after making just nine and, after Azhar and Babar Azam batted through the remainder of the morning, Azam edged the last ball before lunch from Josh Hazlewood to give Steve Smith his second catch.
Misbah-ul-Haq scored 11 off 13 balls with a four and a six before being brilliantly caught at short leg by Nic Maddinson off Bird.
Bird bowled Younus Khan off an inside edge to end a third-wicket stand of 51.
The wicket came shortly before the scheduled tea break and the weather meant that interval came early and play never resumed.

The 27-year-old still had two years left on his existing deal but he has now committed himself to New Road for another season.
He is the ninth player in five months to sign a new deal with Worcestershire, who will be back playing in County Championship Division Two in 2016.
"It was very pleasing to see the other contract extensions," said Shantry.
"It shows the level of morale at the club, even though we had a difficult year in Championship cricket. We've got a group of players here that are only going to improve."
Shantry, Ed Barnard, Joe Clarke, Brett D'Oliveira, Tom Fell, Joe Leach and Charlie Morris have all signed for three years, England Lions man Ross Whiteley has signed for four years and Tom Kohler-Cadmore has agreed a two-year deal.
Southport chairman Charlie Clapham has stood down with immediate effect, along with vice-chairman Sam Shrouder following National League relegation.
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Azhar Ali's unbeaten 66 helped Pakistan to reach 142-4 against Australia on a rain-affected first day of the Boxing Day Test in Melbourne.
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Worcestershire left-armer Jack Shantry has signed a contract extension with the relegated county until 2018.