Hi!
First of all, I know: my bid is a little higher than your budget would allow, but I hope you'll find that my experience in this particular task is worth it.
If with "announcements" you mean "channels", then you're right: Telegram API won't work.
That's what I would do (and what I already did in a similar situation): a simple Python application running with Selenium (a framework normally used for testing via browser automation) on an open browser (Chrome works just fine).
The application will simply impersonate the human user using the browser GUI, you would only have to login with the desired Telegram account at each boot of the application.
The selection of the desired group(s), of course, would be a one-time operation.
Now, if you'd happen to need something less invasive, I could make it all work in a "headless" way: you wouldn't see anything on your desktop! The program would simply use a specific browser lacking the graphical interface.
At that point, if you wanted better reliability and up-time, the whole package could easily run in a cloud service of your choice - or in one provided by me.
I understand that there could be suspicion. After all, this would be my first experience here on freelancer, and you haven't any means to evaluate my actual experience on the matter.
Still, as you can probably see, I feel pretty confident about it.
Don't hesitate to contact me for further information!
Regards,
Davide