Okay, there are five things you can do here. None of them will be what you're looking for, so I'm afraid you'll have to settle for 'second best'.
1: Cut out a mask that much more closely fits the shape of the figure you're trying to dodge. Then place that on the paper, or hold it just above the print, to dodge that area. You'll still have some dark (inside the figure) or light (around it) halos, unless you make an extremely precise mask and sandwich it with the actual negative. Which brings me to...
2: Supplemental masking in order to reduce overall contrast of the negative. This in principle will affect the entire image, making it more muted. Also count on practicing a couple of weeks or months and spending a couple of hundred $$$ on materials to figure out how masking works.
3: Dodge like you're doing now, but do a lot less of it, and instead, create tone in the background by flashing the paper. This will create a more muted, light background while still retaining detail in the subject. Contrast in the background/sky will be reduced.
4: Accept the print as it is, either with your heavy dodging, or the straight print with the dark subject.
5: Re-shoot the scene, this time balancing the exposure of the subject using e.g. fill-flash, choosing a different time of day, using a reflector etc.
Heavy-handed dodging and burning tends to be (painfully) visible. There's little you can do to prevent that. In this case, I'd try to dodge just the face (with a very small mask/implement) and maybe the white sections on the horse to give them a little more sparkle.