You're Correct!

A connected golf swing will lead a golfer to consistency and control throughout all the clubs in the bag. Many golfers find they are swinging a lot with the arms and not enough with the shoulders, hip and core muscles of the body, leading to irregular strikes and directional issues.

Swinging the golf club and predominately using the arms to do so will encourage a poor swing path, generally out to in, which will create a sliced shot and loss of distance. A connected swing allows the player to rotate keeping the club on plane and creates enough tension and power through the golf swing to hit the ball a long way.

To work on this swing change at the driving range, focus on these key areas:

A solid set up routine including posture and width of stance
Takeaway keeping your triceps close to the body and armpits closed
The right elbow should be closed at the top of the swing with the right armpit totally closed. Make sure the swing has rotated using the shoulders to get to this point.
Both arms remain close to the body on the rotation through the down swing and essentially through impact.

The best drill to use when trying to implement this change after your posture and stance is to get a towel and put it across your chest and under your armpits. This will keep the armpits closed to keep the towel in whilst you swing. Use this towel for the majority of your first practise session. This should then give you a very connected feeling in the swing.

This swing change should be done on the practise tee because the timing of the shots will suffer to start with.

When a player uses the arms to push away on the takeaway, this will create a less connected swing because the arms are now too far away from the body. As a consequence, the control of the club and swing will suffer causing poor swing paths and ball striking.

Keeping the arms close to the body during the swing is totally correct to create a connected swing but sacrificing the shoulder rotation will only sacrifice the power that can be created in the swing. Make sure that the shoulders and hips rotate to enable the arms to stay close during the swing.

There is certainly a limit on how connected the swing should be. The hands and arms will stay close to the body as it rotates, however, if they are too close they will cause a flat swing path. For example, if the hands come in too close to the right leg on the back swing, this will encourage an out to in down swing path which will result in pull or sliced shots.