I strenuously agree...(sorry, long)that grandma's emotional instability is contributing to ds' difficulties. His grandmother is using him as her crutch - her 'happiness' - to get through the day. If her life revolves around your son, literally, what is going to happen when Ds begins to develop a life of his own, wants to explore new things, go new places, and isn't available to spend the assigned days with his grandmother? It's time to nip this unhealthy situation before it becomes a bigger problem than it clearly already is.
What is going to happen when ds is 4? 5? 7? 10? He is definitely not going to want to spend all day, days at a time with his grandmother.
Ds should be cultivating his own interests, discovering what HE likes to do, making new friends, playing with kids his own age, and so forth. His role in life is not to fulfill his grandmother's need to have something to do each day. It is unfair and unhealthy, and unreasonable for her to expect him to spend assigned days with her - this affects the lifestyle and health of your family.
If Ds has begun to display behavioral issues such as stuttering that weren't present before, this means the problem has already begun to seriously affect him. Stop the scheduled visitation now - by no means do I mean he should never see his grandmother again, but all visits should have limits:
*No overnight stays
*No all-day stays
*No unsupervised visits (meaning you don't leave him alone with her)
*Limit the time you spend at her home to a few hours per visit
*No 'assigned' days to come over. Keep things flexible.
Of course, this isn't going to just come off without a hitch - if this has been going on for some time, of course there will be lots of fussing, kicking, bag-packing for guilt trips, yelling, tears - you get the picture. What is paramount is that you and Dh stay calm, assure grandma that Ds is NOT being removed from her life, but that as a growing, active child, assigned days to visit and all-day visits are no longer feasible.
What is at stake is the well-being of your son. He is being used by his grandmother as her purpose and direction in life, apparently, and this can be highly damaging and unfair to him- his job is not to provide his grandmother with companionship and a reason for getting up each day - his job is to be a kid, have fun, grow, explore, and do new things.
Let's look at it this way: You see her negative behavior when you're with her - and it disturbs you - but as an adult, you have the option to leave. He doesn't. He has to endure whatever negative behaviors she displays from moment to moment, and it's apparent her behavior is causing him significant stress and turmoil.
Discuss this with your husband and let him know that you can plainly see the current situation is unhealthy for Ds, and that it is going to stop effective immediately.
Feel free to use any of the reasoning I outlined above, if it appeals to you.
Begin by stopping any overnight visits. Then, begin curtailing the all-day visits and don't leave him unsupervised with her. I don't mean to imply that she is deliberately harming him in some way, but if he didn't exhibit stuttering before and is doing so now, you have to consider the possibility that she may inadvertently be harming him in some way through behavior.
Next, don't give in to bringing him over on 'assigned' days. Allow her supervised visits when it is convenient for you. There is nothing she should be doing that you can't be there to view.
Don't leave her spinning, though. Have a plan in mind to encourage her to cultivate other interests so that Dh will be appeased that he isn't abandoning his mother. Did she ever express an interest in travel? Gardening? Is she a vocalist, a pianist, does she like cross-stitch? Help her get interested or re-interested in something new or something she may have once liked to do. Give her information on club memberships, like a cross-stitch club. Give her some gardening apparatus for her birthday if she's into that, give her information on gourmet cooking classes if she's into cooking.
The point is, while taking the appropriate steps to remove her unhealthy dependence upon your son, you can take some relatively easy steps to direct her focus elsewhere to more healthy interests.
This should be prime time in her life to cultivate new things - her kids are grown, and she should be able to enjoy visits from her grandchildren without wanting to be their mother. Perhaps she's experiencing somewhat of a 'life crisis', in that she's no longer a mother in the strictest traditional sense - by that I mean she's no longer actively raising children, i.e, feeding, housing, bathing ,clothing, nursing - you get the picture.
She's still a mother, of course, but her kids don't "need" her to tend to their physical needs anymore. Perhaps that is the most important aspect of mothering to her, and losing that - and seeing the chance to get it back through grandchildren - is causing problems for her.
Sorry this became so long - but your son's happiness and well-being is at stake. End her dependence upon your son now and steer her gently towards her developing other long-term interests. But your son comes first, and it will be a "one-day-at-a-time" situation, where you will need to decrease the dependence gradually while immediately removing your son from this situation.
I would also suggest - without implying that there is something 'wrong' with Ds - that you might have Ds visit a child therapist to determine the cause of his stuttering, and to help eliminate it. The stress may be too much for his young mind, and he's coping in this way. A good child therapist should be able to help him cope with this and overcome the stuttering.
Best of luck to you.