The door lock will lock when I pull down on it when I close it

I would replace it, for safety and peace of mind.

Remove the guide plate first then unlock the door. Take a old lipstick and cover the locks shaft on the end that goes into the plate. Hold the lock or knob open so it doesn't smear on your door faceing . When you shut the door. Then rotate the lock a few times to Mark where it hits the door faceing . That is where you need to drill or chisel out the wood deep enough for the lock to fully engage . Then just put your guide plate back on over your new hole . You can do this your self . I am a 55yr old woman and did it I'm sure you can.

Before you do anything open door lock the door & check to see if lock is working. If it is then do what 7172patches said.

I'm a locksmith. First, check the lock. It sounds like it should be replaced.

Check the hinges, the screws could need tightening. If they won't tighten you will need to take the hinge off and fill it toothpicks or match sticks and woodnglie. Once it dried screw it back in. This is the problem a lot of the times. If they are tight then check the door jam and strick plate. You may need to take a chisel and lower the strike plate. Be very careful not to chisel out to much. If needed you may have use the toothpicks again to fill the screw holes.

It is a doable homeowner project.

I agree with replacing it. Your families safety is most important.