Directors of the National Bank of Commerce have elected David Bindeman, a real estate executive and a member of the bank's board, as its chairman.

Bindeman replaces Jeffrey N. Cohen, who resigned the post to work full time on a real estate project in the Shaw neighborhood of downtown Washington. Cohen will retain his title of chairman of the bank's holding company, Commerce Bancorp Inc.

Cohen voiced his approval of the appointment, saying that Bindeman fulfills the requirements for the job "in every way," and would "revitalize" the bank and its staff.

A group of investors led by Cohen, and including Bindeman, acquired the Washington bank four years ago. Although still relatively small compared with other Washington banks, National Bank of Commerce has since almost doubled its assets, from $42 million in 1980 to $78 million this year, and dramatically increased its profits. The bank recently announced a 112 percent increase in profits for fiscal 1984 over the previous year.

Bindeman, 41, whose only banking experience has been on the boards of National Bank of Commerce and the Prince William Bank in Virginia, said he regards the job of chairman as one of bringing the bank new business and of providing guidance for the bank's management.

Since 1980, Bindeman has served on the bank's audit committee, its loan review committee and has been a member of the bank's executive committee. In addition to the time he plans to spend at National Bank of Commerce -- a minimum of two to three days a week, he said -- Bindeman will continue his work as president of the Landmark Cos., a family-owned real estate company.