The main thing with weighted balls in my opinion is to have a firm foundation of throwing in place. Throwing mechanics (not talk about pitching here, but throwing) should be solid as well.
I would do an “on ramping” into weighted balls if you decide to do them.
Driveline baseball has an excellent free ebook you can get.
The underweight balls are actually more dangerous than the over weight balls. Arm speed is what is a main driver in injury (a majority of injuries occur at or after ball release and the arm attempts to slow itself). There is a reason football QBs rarely have elbow injuries and baseball players do. The football is so much heavier than a baseball the arm is moving slower.
Find a good program, take your time and be very careful introducing underweight balls.