Can someone access my Messages with just my password?

Joycecarr wrote:

I was told that anyone on my account with the password and open my texts to read them, any truth to that?

In order to do that, they would need to log into your Apple account on an Apple device. It can't be done through a browser. To prevent this, enable Two-Factor Authentication on your account:

Two-factor authentication for Apple Account - Apple Support

yes, anyone knowing your Apple ID and password has access to most of your data, not only to your messages