The D5600 is a capable camera for bird photography, so a body upgrade is not automatically necessary. In most cases, the bigger factor is your distance to the bird rather than the camera body itself.

Your current 300mm on the D5600 already gives a field of view similar to about 450mm on full frame, which is decent reach. If you still can’t fill the frame, a 400mm, 500mm, or 600mm lens can help, but those lenses are larger, heavier, more expensive, and harder to aim quickly.

So the main tradeoff is reach versus size/weight/handling. Before upgrading the body, it makes sense to keep using the D5600 and learn whether you truly need more reach or whether fieldcraft and getting closer would help more.

If you do want more reach, third-party Nikon-mount telephoto lenses are worth checking, including Sigma options. Another possible route is using a teleconverter with a compatible telephoto lens, though that depends on the specific lens and converter.

In short: keep the D5600 for now, and consider a longer Nikon-mount or third-party telephoto only if your shooting distance consistently demands it.