The 62-year-old has lost almost three stone in recent months and was urged to visit doctors by his wife Catherine and other family members.
Sir Lindsay said the symptoms were so severe that doctors wanted him to stay in hospital, but he refused to miss the election campaign.
He is now receiving treatment for what is most likely to be Type 1 diabetes and awaits further tests.
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Sir Lindsay said the diagnoses "is still a shock" and his family are "very worried", but said he is determined to continue his roles as an MP for the Lancashire seat of Chorley and as the new Speaker.
"I'm on tablets, as well as having to inject insulin, but it doesn't stop me carrying on and nothing is going to be a barrier to me," he said.
"I'm going to cope with it. I'm going to manage it. I'm going to get through this.
"The fact is I feel really well. We know what it is - that's the good news - and of course, I have got to get over it and get on with my job.
"The House of Commons elected me to be the Speaker and there's nothing that's going to stop me from doing that."
Sir Lindsay cited former Prime Minister Theresa May, who also has Type 1 diabetes, as an inspiration and says family and Commons' staff have been very supportive.
Sir Lindsay spoke about his health condition during an interview with Rob McLoughlin for the forthcoming series Mr Speaker.
Chris Askew, chief executive of Diabetes UK, said: "Living with Type 1 diabetes can be hard, but as Sir Lindsay's experiences have shown, with the right support from your healthcare team - and careful management - people can live full and healthy lives following their diagnosis.
"It's often thought that Type 1 diabetes only affects children but, while it's less common to see someone of Sir Lindsay's age diagnosed, it can affect a person at any time in their life.
"That's why knowing the signs and symptoms of diabetes - the four Ts - can be a life-saver.
"So if you're going to the toilet a lot, experiencing increased thirst, are more tired than usual, or losing weight without trying, you should speak to a healthcare professional."