If I had a sentence I had my own room at my grandparents' place. I think I would place the apostrophe as shown because I'm talking about both grandmother and grandfather. However, let's say I shorten the sentence to I had my own room at my grandparents'. Am I correct in leaving the apostrophe as I have done?
Yes, you're correct.
Your second sentence is the first sentence but just not saying the final word.
In cases with unspoken words, the forms of the remaining words generally remain the same as if the unspoken words were there.