Welcome to 2CarPros.
The battery shouldn't be causing this issue. However, there are a few things I suspect. The flashing of the D on the dash could be related to an issue with what is called the range sensor. And the stability light could be related to a wheel speed sensor issue.
Notice that I said "could be". What you need to do is have the computer scanned to identify diagnostic trouble codes. Since the engine light is on, there will be codes, and they should point us in the right directions.
Here is a quick video showing how a scanner is used:
https://youtu.be/YV3TRZwer8k
Now, I realize most people don't own a scanner. However, often times a parts store will do it for free, or lend or rent one to you.
That is what needs to be done. Otherwise, it becomes a guessing game. As I mentioned, a new battery shouldn't have caused this. It is something new. If you can get me the trouble codes, I will be able to better direct you.
Let me know.
Joe
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Thursday, April 18th, 2019 AT 9:00 PM