In your example "wobble" means he hesitated or wavered in his decision to sends troops in.

It means that when something is not standing steadily, and it can fall down, or wobble over. For example: A baby can wobble over when sitting. A vase can wobble over. A stuffed animal with a big head can wobble over because its big head is pulling the whole thing down. Example sentences: The pile of bricks wobbled, then fell. My baby is trying to sit on her own but yesterday she always wobbles over. It is very funny :) I hope I could help.

Wobble over as used in your example is not a phrasal verb like fall over or topple over. Susan's answer is correct.

waffle is a more common term for hesitating in making a decision. Someone probably misheard waffle and assumed wobble because they sound similar.

The President appeared to wobble over sending the troops in.