Is there a website or program that.....

I want to actively promote music to radio stations and I want to keep texting request to the radio stations via their short 5 digit text number!

Does
ANYONE know of a program where I can send my text messages on a timed schedule? Alternates the sending phone #?

I am not sure if this will help you but here goes:

I use Voipstunt for my calls, it provides a scheduled texting service from within their program. I think you can use their online service for this too but not sure.
To have some number attached to that you can use a number that lets you answer the phone without voicemail.

Voipstunt is part of a bigger German company of which I forgot the name but they have multiple websites out there.
As an extra benefit you would have 120 days of 3 hours a week free phone calls to most landlines all over the world which could be beneficial in other ways.

This does not alternate but it seems bots like to do this for you. To write one shouldn't take to long I think. I won't write it for you though.

Just a thought, ignore as you see fit.

Try request it on freelancer