In a tweet, Mr Adams said: “Yesterday, Selby Conservatives selected an excellent new parliamentary candidate.
“I’ve today informed the chief whip that I will be standing down as a Member of Parliament with immediate effect.
“It has been an honour to represent the area where I was raised (and) educated.”
Mr Adams and Ms Dorries had been tipped for peerages in Mr Johnson's resignation honours but neither featured in Friday's published list.
There has been speculation that Downing Street wanted their names removed to avoid by-election contests, although No10 said Mr Sunak had no involvement in producing the final list.
The Prime Minister has yet to comment on Mr Johnson's departure.
While the Tories enjoy a comfortable 20,000 majority in Selby and Ainsty, Mr Adams' decision to exit immediately rather than wait until the next election means the Tories will face contests on three fronts.
Mr Johnson sensationally quit Westminster on Friday as he launched a fierce attack on the Commons Privileges Committee investigation into whether he misled MPs with his assurances over parties held in Downing Street during coronavirus lockdowns.
In a 1,000-word statement, he said the seven-person panel, which is chaired by veteran Labour MP Harriet Harman but has a Conservative majority, was on a "witch hunt" and compared it with a "kangaroo court".
Mr Johnson said he was "bewildered and appalled" at being "forced out, anti-democratically" by a probe that he claimed had set out from the beginning to "find me guilty, regardless of the facts".
He denied lying to MPs and said he "corrected the record as soon as possible" after receiving information about lockdown gatherings in No 10.
The former Tory leader's Uxbridge and South Ruislip seat was in Labour's top 100 targets at the next election even before Mr Johnson's resignation.
The Liberal Democrats have briefed that they could potentially pull off an upset in Ms Dorries' former Mid Bedfordshire constituency.
The TalkTV presenter said on Friday that she did not plan on triggering a by-election but resigned shortly before Mr Johnson's honours roll was made public.
Labour came second in Selby at the 2019 election.
As well as the sudden departures, Tory Will Quince and Sir Bill Cash both added their names to the growing lists of Tory MPs who will step down at the next election.