좋은 질문이에요!
Great question
아니다싶어's "싶어" or "싶다" means "to think"
So it's the same as 아닌 것 같아. or 아니라는 생각이 들어.
You can use -다 + 싶다 when you want to say something like,
"looking at the circumstances, I feel that is is .... "
But it's most commonly used in the form "아니다싶어", but if I TRY HARD to make some sample sentences, I could say things like
이거다싶었어요. (I thought this was the right one to choose.)
이 사람이다싶어서 결혼했어요. (I thought she was the one for me, so I married her.)
이건 아니다싶어서 생각을 바꿨어요. (I thought it wasn't right, so I changed my mind.)
I hope this helps!