You're Correct!
When taking up your grip on the golf club handle, it is really important to get your hands working as one unit, rather than two separate units.
The easiest way to achieve this is to make sure that your hands are connected, by either interlocking your grip or overlapping.
An interlocking grip sees a right handed player interlocking their left index finger with their right little finger. An overlapping grip sees the little finger on the right hand sit between the left index finger and middle finger groove. However, it is the position of the hands, rather than their connection that is crucial. Provided that your hands are positioned correctly on the handle, if using a baseball grip, which sees the hands touching each other so that the right little finger sits on the handle next to the left index finger, easing any finger pain that you are experiencing, then it will not have a detrimental effect on your game.
To position your hands correctly on the handle, you need to point the fingers on your left hand directly down towards the ground. Place the handle of the club across your hand so that it is positioned from the middle joint of your index finger to just below the callous pad of your little finger. Close your fingers around the back of the handle and place you thumb on top. The key to the correct grip is to be able to hold a coin between your thumb and index finger so there is no gap. This coin should be positioned on the front of the handle and so that it points up to your right shoulder.
To place your right hand on the handle once again, point your fingers directly downwards at the floor and position the handle from the middle joint of your index finger to the callous pad of you little finger. Position your right hand lower down on the handle than your left. Wrap your fingers around the back and place your thumb on top. Now slide your right hand up towards your left so that it sits on top of your left thumb. Allow the scooped out part of your right palm to locate on your left thumb and you should be able to hold a coin between your thumb and hand, which points up to your right shoulder.
Hand position is crucial as it gives you directional control over your shots as it controls the club face position through impact. If your hands are not positioned correctly on the handle, they will rotate into the correct position through impact and as a result alter the club face position and affect the ball flight direction.
But whether you baseball, interlock or overlap should not matter. If having a baseball grip where your hands touch each other relieves finger pain, then use this when you play.
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If your fingers are hurting when you are playing golf, then gripping the club even tighter than you are currently doing will not help to alleviate the problem. Gripping tighter will make your fingers hurt even more so try relaxing your grip pressure slightly as this should help.
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If you interlock your right little finger and left index finger (right handed golfer) and you interlock too deeply, this can cause pain in your hands. If you interlock with your grip, you need to make sure that you do this so that the webbing in between your fingers on opposite hands does not touch. If it does, you are interlocking too deeply and it will cause you pain in your fingers and will also give you further issues in your swing. Interlock less deeply for more comfort.
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If you split your hands apart on the golf club handle, you will find that your hands begin to act independently of each other rather than as one unit as they should. Keep your hands together and if you have no other choice than to baseball grip, keep your hands touching each other, do not let them split apart.