Hi Danielle, I feel for you. This is a tough situation. I guess the good news is that your grandmother is wearing diapers (I recently had a question from someone
whose parent refuses to do this, soiling herself.)
I think it's time for some tough love. I know this won't be easy, but you need to insist that your grandmother regularly change her diaper (at least daily, if not more), and dispose of her used diapers. The smell is one thing. But by wearing a damp diaper, your grandmother is at risk for skin conditions and infection.
I think the easiest approach is to consult with your grandmother's doctor, and ask him or her to have a talk with her. The doctor can explain the reasons regular diaper changes are important. If your grandmother refuses to see her doctor, maybe the doctor -- or a nurse -- is willing to chat with her by phone. A second idea is to hire a home health nurse for a one-time visit and consultation. The nurse can talk with your grandmother about managing her incontinence and include information on diaper changing and disposal. If the doctor or nurse approach this as helping your grandmother manage her health and well-being; she'll be more likely to listen. (They can also explain to her how our sense of smell often weakens with age, so she may be smelling the urine, but the odor is still there.)
It's also helpful to remember that in many cases seniors would rather discuss personal or intimate issues such as incontinence with professionals or hired staff than family.
Finally, if your grandmother thinks diapers and garbage bags are a splurge, I wonder if it would help if you bought a huge supply at a warehouse store like Costco and emphasized what a great deal you got, so she doesn't need to worry about the economics of the situation. Maybe you've already tried this kind of thing.
Best of luck with this tough situation --- and with your new baby as well!