NFHS rules only.
R1 on 3rd, R2 on 2nd, R3 on first, no outs, batter runner hits ground ball to F5 who steps on the third base for the force out. As BU, shading F6 towards third, this was an easy play since R2 was out by 10 steps, I was five steps from third and only signaled her out. The third base coach told his girl to get back on third base (she was walking off the field knowing she was out) and the defensive coach started yelling from the other side of the field for me to call the out. What is the accepted mechanics here? Does the force out have to be called?

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Seems adequate to me, if you use the proper mechanics of showing
the runner was out, why yell, "runner's out"

On the front end of a double play, I will usually only give a verbal and not a signal. If it's a banger, then both. Especially when the runner is out by 10 feet, my verbal is acknowledging that I saw it & called it. With the forced runner out by 10 feet, everyone involved knows it - just acknowledge the out. While I'll then begin to move toward the next play, my eyes will stay with & on the ball until it's released.

On double plays, I always call the out at the front end of the play as well. The mechanic WW originally posted is acceptable, however it is not preferred. The first call of out should be both verbal and with a signal, the second call should be a sell safe or out!