Generally speaking women have a 28 day cycle in which you ovulate (generally) about 12-14 days after the first day of your period. And a "window" of about 2 -3 days to become pregnant around the time of ovulation. If you have a "normal" cycle of 28 days (not everyone does!) you would typically ovulate (have the 2-3 day window where you can become pregnant) around the 14th day of your cycle. But since some women's cycles are often longer (or shorter) than 28 days, the "window" can be closer to the 16th or 19th days or so. Basically, it will be about halfway through your cycle. If your cycle is normally 32 days, then your window would probably be around day #16 (not day 14). If your cycle is 38 days, your window would be around day #19.
When I was trying to conceive with our second child, my cycle varied widely from 24 days to as many as almost 50 days!
Once you understand that, then you just use that date and count forward 40 weeks. A pregnancy is NOT 9 months. It is actually a bit closer to 10 months.
So if your last period began on May 29, and you have a "normal" cycle of about 28 days, you would count 14 days into that cycle (which would be June 12) and then from that date count 40 weeks. That would estimate your due date to be March 19. If you have a longer cycle then you might need to move the conception date back a few days more. And keep in mind, that illness, stress or other factors can influence when you actually ovulate. So if you adjust the date a week or two for stress or a longer menstrual cycle, then you could easily have an April 1 due date.
It seems highly improbable that you started your period on May 29 and did not become pregnant until July 10. That would mean that you didn't ovulate until around day 42 of your cycle! It seems EXTREMELY more likely that you conceived closer to day 27 of your cycle.
And the April 1 due date you were given by your doctor EXACTLY corresponds with 40 weeks from June 25.
If you still have concerns then you should discuss them with your doctor. He/she must maintain patient confidentiality and not disclose your concerns with others. It is important for you and your baby's health to not be so stressed out over this, and information is what you need to allay your fears. Have your doctor to explain to you exactly how your due date (which is just the reverse of saying date of conception) was determined. Was it strictly by the possible dates of conception you gave him? From the date of your last period? From measuring your uterus? Or some combination of all three? If the baby measures differently than what they would expect judging from the date of conception, then that should be brought to your attention, and I'm sure will be as it can relate to health issues of the baby.
But please talk to your doctor and get them to explain in detail how they arrived at your date of conception. It is important (not just for the reasons you are after at the moment) but for your and your babies HEALTH.