Combine the lime and orange juices, rum, garlic, lime and orange zests, vinegar and sugar in a blender and process until smooth.
Prick the chicken all over with the tines of a fork and rub generously with salt and pepper. Place in a large bowl and toss with the lime-orange marinade. Cover with plastic wrap and refrigerate for 6 hours or overnight.
Preheat the oven to 425 degrees. Place the chicken in a large roasting pan, reserving the marinade. Brush the chicken with oil and bake for 15 minutes until it begins to brown.
Reduce the oven temperature to 350 degrees and bake another 10 minutes.
Remove the chicken from the oven and add the papaya and the reserved marinade. Return to the oven and bake, basting with the marinade until the chicken is falling off the bone, about 30 minutes.
Transfer the chicken to a serving platter. Taste the sauce and add salt if needed. Spoon the sauce and papaya over the chicken and serve.