Both are correct. The two choices differ grammatically and emphasize different things. Neither is better than the other.
"We felt something move" uses a "bare infinitive". In addition to "feel", the verbs "see", "hear", "make", and "let" are often followed by a bare infinitive:
"We heard her sing"
"They made me dance"
"Let me play"
"We felt something moving" uses the adjective "moving" to describe "something". In other words we felt something and it was a moving thing.
The difference in emphasis comes from the fact that adjectives are descriptive. "Moving" is an adjective that paints a portrait to help you to imagine what was happening.
The first choice uses verbs alone. This places the emphasis on action, not description. It doesn't paint a picture. It simply tells what happened.