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From what you have told me, it does appear that Lucy has a mild digestive disturbance if she is still eating well. The slime you see if mucous that covers the stool that has been in the colon too long because Lucy is too painful to release this. the green is due to an excess amount of bile being released. If this is minor and due to something Lucy ate or stress, the best thing to do for now is to fast her for 24 hrs. After the fast, feed a meal of boiled chicken ¼ and rice ¾. If this has helped, wean your dog onto the regular meal over 2 day’s period. At the same time, you can give Imodium AD 2 mg caplets at 1 caplet per 30 pounds of Lucy’s weight given every 8 hrs.
It is extremely important to have Lucy seen immediately by a vet if she continues to have severe diarrhea. Even the dehydration from this can cause her severe complications.
If this is not successful, you will have to have your vet examine Lucy to determine if there is something else such as a protozoa or infection that is causing this diarrhea.