Wider straps can help distribute the weight, resulting in less digging-in. One solution commonly seen are racer-back bras without hooks, they just slip on over the head. The problem then becomes how to get the wet, sweat-soaked bra off after a good walk. If you are prone to sweating while walking, look for a wicking fabric such as CoolMax or polypropylene in a sports bra. This gives you the versatility to take your shirt off if you overheat.
Most sports bras come in a variety of colors and patterns.
Check for whether the fabric is opaque even when wet and if the fabric pattern or texture hides the nipples enough for your personal modesty. It takes more than spandex to control motion for larger cup sizes. Motion control is evolving but many readers still could not find the perfect motion control bra. Where there is sweat and motion, there is chafing. A sports bra can prevent chafing by reducing the rubbing.