Hello, Danielle, and welcome to TP. I hope you will find good advice and support here.
The Alzheimer's UK Society has a lot of very good information on their website. Here is what they suggest about communciation:
https://www.alzheimers.org.uk/site/scripts/documents_info.php?documentID=130
There is also a very good compassionate communication leaflet. While this is not exactly what you're talking about, it might be of help to you:
https://www.alz.org/co/documents/communication_strategies.pdf
(Please note the phone number on that one is a US phone number.)
And this article from the Aussies:
https://fightdementia.org.au/national/support-and-services/carers/managing-changes-in-communication
My mother is 74 and has Alzheimer's (at least moderately advanced) and no short-term memory. As you have noticed with your grandmother, there is an erosion of personality and conversations are different now.
My mother does best when she is looking directly at the person who is talking to her and can make eye contact with them. Sometimes she loses the thread of a conversation but can still understand body language and tone of voice, so those are important. She does not like to be questioned, so instead of saying, "what was that story you used to tell me about Uncle Jim and the car?," I will say, "I was thinking about Uncle Jim the other day and that story about him and the car." Then she may chime in with a reminiscence, or not, but doesn't feel pressured.
I've also noticed that conversation with my mother needs to be less abstract and much more direct and concise now. A long sentence or something complicated, can just confuse or even upset her.
I think some people find that making a memory book or photo album and going through it together can spark conversation.
Your grandmother may, or may not, have times of the day when she is more chatty or more open to conversation. This is usually not late afternoon and evening, but you might experiment. Also, some days are better than others.
Patience and compassion go a long way. If your grandmother is not distressed by not finishing a sentence, then you might try to just run with it (easy to say, hard to do). Unless it's important for her to tell you something (information about pain or distress), maybe it doesn't matter if she finishes her sentences. Sometimes all we can do with dementia, is manage our expectations.
Best wishes to you.