When you’re playing out of wet, compact sand, instead of digging that lead edge into the sand, try what I like to call a “putt chip” where you do something similar to your putting stroke with your wedge.
Using your putting grip, put the ball back in your stance and stand a little closer to the ball, which will help you to hit the ball off the toe of the club. Taking a longer putting stroke, you’ll pick the ball cleanly, making it pop out and have a little check on it.