My advice to you would be to start the job and when you are making the news public among your family and friends you can announce it at work.
If you wished, you could always speak with your direct manager confidentially sooner than that and advise him/her , just to give them as long as possible to consider their long term cover plans.
There is nothing to be gained by telling them in advance of signing the contract. Plus, your being pregnant has absolutely NO bearing on your ability to do the job, or on the validity of the job offer. It also reeks a little of "confessing" to some misdeameanour, and that you want to "come clean" before you start the job. That attitude just does not sit well with me at all. There is NOTHING shameful, or wrong, or inconsiderate, or anything else negative about your being pregnant. You have EVERY right to be pregnant and to plan your family according to the times that suit you and your family.
I would advise you to keep this in mind when telling your employer your news. I would further advise you to ensure that when you are telling your employer that your body language or your chosen words do not in any way convey some kind of apologies for being pregnant... again, by using words like "I'm sorry to tell you...", or "I feel bad saying this...." or "I wanted to tell you before I started...." etc etc give the impression of wrong doing where there has been no wrong doing. The language you use in your post (
go into a new role being dishonest...I would rather not lie...) gives the impression that you do see something wrong with being pregnant, and I can't for the life of me see why this is.
How have you lied, or how are you being dishonest... did someone ask you outright if you are pregnant and you said no? By keeping your private news private, how is this lying? Any manager worth his/her salt knows that if they hire a woman of child bearing age, then there is a possibility that she might get pregnant, or be pregnant. They cannot exactly be bowled over by such news, and further, 4 to 5 months is PLENTY time for a manager to arrange cover.... after all, if someone is leaving a job they typically give 1-2 months notice and businesses around the country manage to carry on, and recruit and backfill roles in time. Don't stress about it.
For what it's worth, I am a HR Manager and also a mother of two smallies, so I write with a business hat on as well as my personal hat on.
FYI......I started this job when I was 11 weeks pregnant. When I told my manager, she and her manager offered me their heartiest congratulations. I passed my 6months probation and went on mat leave within couple of days of one another. I came back from mat leave, was promoted 2months later and then 2 months after that I told them I was pregnant again. When I came back from my second mat leave I was promoted again within 3mths. In other words, there were no negative reactions. Hopefully you will experience the same modern/enlightened response.