If you spoke with your doctor explicitly about the need to know when you conceived (and didn't just say "How far along am I?"); in other words, if the doctor knew exactly what you were worried about and gave you that time range for conception, I'd trust the doctor's information.
If you would like, you could give the information like this (filling in the blanks with calendar dates, not "weeks" of pregnancy), and we could count it out just to reconfirm what the doctor said.
First day of last period: February ____
Sex with Guy 1: February ___ through March ___
Sex with Guy 2: February 8
First ultrasound: ___________________________
(Doctor said date of conception was Feb. 25-March 1)
I think the doctor is right and the ultrasounds are right and that you really don't have anything to worry about. It sounds like your two kids are from the same dad.
Do you live in the United States? Things are pretty cut-and-dried here regarding getting a DNA test done that is legally acceptable. Don't just go to the drugstore and get an off-the-shelf test ... if you ever had to prove he was the dad, the courts would not accept those results, and you don't save much money over doing it properly at the hospital when the baby is born, anyway.
I wouldn't let his fantasy of a big white wedding stop you from getting married, with two kids in existence who need certainty of their parents' status more than they need you two to blow a bunch of money. You can always have a big celebration in a year or two, there is no law against having big parties any time. But there is nothing great about a big white wedding if a couple has other more serious places to put their money, like, a down payment on a house or getting the family on a firm financial footing.
The reason I said to do a DNA test with your first child when you do one with the second child is that you will need it in the (unexpected) event that the second child is not from him. If your boyfriend leaves due to a surprise in the paternity test of your second child, you *will* be in child-support territory and will need all the legal help you can get. The resemblance between father and son is not evidence that will matter to a judge. If you ever had to go to court to get child support for that child, you would need DNA evidence. You and your boyfriend need to get serious, either get married or you should get the legals in place in case he decides to take a hike. Sorry to sound brusque, but you right now are in a position where you could be totally left holding the bag and you sound very vague for someone who is walking a tightrope over boiling lava.