To answer your main question: yes, you can save an animation as a gif from Photoshop CC. You won't need to rebuild it in Flash.
To echo others, it sounds like you might have some misconfigured settings somewhere or possibly your animation isn't built properly. Posting more detail would help. I've tried to reproduce your problem and can't seem to come up with what you've described. In the meantime, here is the process for creating a GIF from Photoshop CC:
First adjust your frame or timeline animation (either type of animation should work, though I prefer the control of frames when making a GIF). Then select
Save for Web (Legacy)…
In the Save for Web dialog, ensure your export, compression, and looping options are set how you want and test the animation playback.
Save and choose your save options. There is no need to select either of the HTML options if you are just exporting an animation to GIF.
because there are several distinct steps involved in creating an animation and exporting it as a .gif file via Photoshop, it's important to test your animation's playback throughout the process.
Testing along the way will not only ensure that you get desired results in you final file, it will also help you know which part of the process to troubleshoot if something turns out wrong.