Car in Drive Acts Like it’s in Park When I Stop

I am posting this here for Mandy.

Please help. I have an older car starting to have transmission issues but the issues are very strange and more inconveniencing and embarrassing than dangerous at this point.

What happens is this: I start the car. The car goes into drive after me shifting back and forth from reverse to drive for about 3-5 mins. I will drive for a while and all will be fine until I come to a complete stop. When I stop, the car will shift out of drive. What I mean is, it will say it is still in drive but act as if it is in park. I hit the gas and I go nowhere. So, I go back and forth from reverse to drive (for another 3-5 mins though this time with cars behind me) until it clicks back into what is actually drive and I can continue forward. This is how I drive my car now.

My questions are:
1. Is it safe to drive it like this? (there have been no problems when the car is moving, only when I stop)
2. I know it’s transmission related because my local shop looked at it but they don’t do transmissions but they didn’t tell me exactly what was wrong. So can someone please explain what is wrong and how much it will cost to fix?

Thank you 🙂