When you were considering getting the first DNA test, did anyone mention to you that if your due date is in the first week of January, you got pregnant in the second week of April? Not in March? Both of your DNA tests were unneeded, because by the dates alone (let alone the use of Plan B) you didn't get pregnant in March.
A lot of times, women who feel guilty about having had sex with someone else will ignore the facts in favor of freaking themselves out about paternity. But whether you are doing this or not, the due date for a baby conceived March 15 is December 6. If you had gotten pregnant March 14/15, your baby would be here already. The fact that you're not presently holding a 10-day-old baby in your arms should make it clear there is nothing to be nervous about.
About whether to get a DNA test after the baby comes, there is a totally different reason this is a great idea. It provides legal protection for the baby.
If you are not married, your boyfriend is not automatically considered by the law to be the father. If he is standing by ready to sign the birth certificate (if that is required in your state), that should be enough. But couples (married or not) do sometimes break up later. A man who was married to the mother at the time her child is born will be on the hook for child support just from being her husband. (There's a baked-in presumption under the law that a husband is the dad, and takes a lot of legal work to get out of it.) But someone who is not a husband would find it easier to claim the baby is not from him and try to get out of the child-support obligation. A DNA test done at the hospital, noted with the results in the baby's medical file, will help prevent that kind of stuff. It's legal safety for the baby's future support.
If you do go that route, try to make your boyfriend understand that it's just routine, it's legal protections for the baby. Don't stray down the road of implying you think you might break up. And even if you are hiding that you had the other sex, don't act guilty and furtive (though why it's bothering you so much I'm not sure, if you were broken up at the time. It was a month before conception, after all.) Almost certainly, if you two aren't married, some of your boyfriend's friends will have already been putting a bug in his ear that he should ask for a DNA test before signing up for a lifetime of commitment to a child. Your position is that it's for the baby's sake, and his position might be relief that he doesn't have to offend you by asking.
Ask the doctor how to go about ordering a DNA test that is legally admissible in the courts for proving paternity, to be done after the baby comes. And be sure to witness your boyfriend doing the swab and handing it to the lab tech. (Some boyfriends have been known to get a look-alike buddy to show up at the lab and do the swab for them, and that kind of mess causes a lot of grief, and even more testing, to unravel.) Then when the test comes, you will always have this legal proof of paternity, and can forget it and never worry about it again.
Congratulations on your upcoming little one!