Yes, there are no rules stopping you from putting away your shield.
On top of answering the question, I just wanted to check your order of events to make sure you understand them.
Turn 1: You may only shove a creature prone with your shield as a Bonus Action using the Shield Master feat after you take the Attack Action (Which may come before or after the ACTUAL attack given when you take that Action). No problems here. To make a Grapple, you must forgo one of your attacks in place of a grapple as per PHB pg. 195
Grappling is not a free action. So make sure you either shove first as a Bonus Action and then attack; or you have the Extra Attack feature so that you may follow your grapple up with an attack in the same turn.
Turn 2: PHB pg. 144 tells us that a shield is carried in one hand, and PHB pg. 146 tells us it takes an action to drop, un-equip, or otherwise put away (or Doff) that shield. I don't see anything stopping you from doing this while maintaining the grapple with a free hand as per the grapple rules. No problems here.
Turn 3: As per PHB pg. 190 you can draw a weapon as part of the same attack action using that weapon, and since the target has 0 speed because it is grappled it has never been able to get up from the Prone condition you knocked it into at the beginning of all this. As long as you are using a 1 handed weapon in your free hand, you are still able to make attacks while maintain a grapple with your free hand. No problems here either.
All in all, as long as you have the required number of attacks, this is a considerably effective strategy at keeping a prone target down and suffering attacks from you and your friends.