Wouldn't that be the same thing as setting my monitors refresh rate to 60 and turning v-sync on?Try to use frame rate limiters, that fix fps of the game by fixed number (25, 30, 60 fps), usually game ported to PC supports this feature under the game settings (ask game developers about this feature).
What would you suggest for limiting fps, like what program?Not the same. Refresh rate - is how often image updates. Rendering fps - is number of new frames you can get in one second. On 144 Hz monitor you possibly can get more contrast image (fast response time, temporal smooth almost gone) when image changes from frame to frame. But really, there is no benefits from 144 Hz for streaming or recording (maybe 120 Hz for stereo mode but I recommend interlaced stereo monitors instead - where vertical resolution reduced...). If image looks good in 60 Hz mode (no flickering), then you can use it and enable V-sync.