Sports bras are my specialist subject
I used to have very large boobs (up to an H at one point) and an now a more sensible E cup, so have tried everything over the years.
First of all, please don't go anywhere near M&S, Sainsburys, Next, etc. These aren't decent sports bras and may feel supportive at first, but aren't a patch on a "proper" sports bra once you're doing serious exercise over longer distances.
If you are really on a budget then Decathlon is your best budget buy, but I still don't find them supportive enough for my cup size.
The best ones IME are Freya and Panache, closely followed by the Shock Absorber Run. Go down a back size in the Freya and Panache ones compared to Shock Absorber, though.
The Panache one, and the Freya underwired, encapsulate the whole boob and stop you bouncing from all sides iyswim. The Shock Absorber Run is a more traditional "strap em in" style, which ime can lead to more chafing over long distances.
I'm a similar size to you, OP, and currently have two of the Freya underwired in a 30E, and the Shock Absorber in a 32DD, which I rotate. I also have a Freya non wired which I don't get on with as well, but it's basically identical to the Shock Absorber Run.