My oldest and my youngest are both like that.
You cannot force independence on a child. It develops. Everyone, and I do mean everyone, told me how my oldest would be ruined if I didn’t leave him with a sitter/daycare/preschool right away. There were horrible predictions about how he would be 15 years old and need me to come with him on dates because he wouldn’t be able to be away from me, etc, etc.
I ignored all the nay-sayers. When my son wanted me I let him have me. I let him know that I would always be available to meet his needs, and he learned to be secure.
Results: last month when he started Kindergarten, he ran off to play without looking back. In fact, he didn’t even say goodbye. He has no difficulty whatsoever. I will do things exactly the same with my youngest.
Do what you feel is right in your heart. Your children are YOUR children. I hope this helps.