OP, I flew cross country twice when I was pregnant with my DD, and I've already done it 1x this pregnancy (and likely will have to again for a funeral in the near future) and while the radiation exposure is discouraging, I don't think it really accomplishes anything to get really worried about it. I've heard some people try to fly at night to limit exposure to the sun's radiation
However, if I were you, and travel for work was frequent, I would look into cutting back for other reasons. You said you are newly pregnant...I HOPE you don't end up with bad nausea, but you might. With my 1st pregnancy mine started around week 6...this time week 5. I actually just flew cross country at 8.5 weeks and I felt SO sick the entire time. The pilot barely ever took down the fasten seatbelt sign so they kept trying to make me sit down when I needed to use the bathroom to throw up not fun.
With my last pregnancy, my flights were in my 2nd and 3rd trimesters, and I was less nauseaus but I was big enough that the seats were SO uncomfortable. I also had to pee every 30 min which is not convenient. (And one of my flights was 1 day after a newsworthy airplane scare so EVERYONE was on super-alert mode and the flight attendants were VERY pushy about making people sit down)
I can see how flying to a tropical destination might help it feel worthwhile, but if your trips are for business, I'm guessing that you will probably be less tired and much more comfortable with fewer trips! Congratulations on your pregnancy.