The rules with movement and the 5' step applies because if you 5' step and then move then you didn't really 5' step you just had continued movement. Other than that you can move equivalent actions and still 5' step in a round.
Rules from the SRD
You can move 5 feet in any round when you don't perform any other kind of movement. Taking this 5-foot step never provokes an attack of opportunity. You can't take more than one 5-foot step in a round, and you can't take a 5-foot step in the same round that you move any distance.
You can take a 5-foot step before, during, or after your other actions in the round.
You can only take a 5-foot-step if your movement isn't hampered by difficult terrain or darkness. Any creature with a speed of 5 feet or less can't take a 5-foot step, since moving even 5 feet requires a move action for such a slow creature.
You may not take a 5-foot step using a form of movement for which you do not have a listed speed.
Notice how it says in the first paragraph "Movement" and not move action, thats because there is a difference. Your DM was mistaken when it comes to the 5' step + move equivalent action ruling