Hi, I am Dr Megan a Uk Based veterinary surgeon. Lots of things can cause scouring in young lambs including infections (bacterial vs viral), internal parasites (worms), vitamin deficiencies, or dietary changes or a lowered immune system or if the feeding equipment/bottles are not sterilised?
While you can technically wean a healthy lamb at 6 weeks of age, it is best to wait until they are around 10-12 weeks of age. It is good that she is eating grass, and at around a week or two they should start to eat small quantities of grass hay. Foods like chaff and pellets, creeper feeding should be introduced in small amounts only after around 3 weeks of age and hard food is a necessity as it is very important for rumen development.
You can alternate electrolyte and milk feeds, making sure not to mix electrolytes and milk together in one feed, or to give them too close together (at least 1 hour apart), as it can delay nutrient absorption and make the sickness/scouring worse.
To help aid recovery, you could mix in a probiotic such as Protexin (helps maintain gut flora balance). If a lamb is lethargic or unwell, mixing in some Nutrigel will help provide energy to encourage them to suckle, as well as essential nutrients. Scourban, Protexin and Nutrigel can be found at some feed/pet stores, online or at some vet clinics.
However, it sounds like your lamb needs to be checked over by a veterinarian as the scouring has been going on for a while and not improving, to rule out other underlying causes which can be more serious such as infections or parasites.
Hope this advice has been helpful for you.