Well this is the somewhat depressing part: interstellar distances just seem so vast one has to wonder if it's ever going to be feasible. This is worsened by the timescales just being hopelessly long for how long we live. It seems far more likely that AIs are more viable to send than living organisms.

We will not live in meat forever. Once strong AI happens, or maybe even before, human consciousness will be transcribable into more resilient matter. That type of consciousness could be hibernated for millennia in transit. Or play endless games in transit. Its a bit silly to imply "living organisms" a being somehow better, especially when the vast majority of future humanity will live their potential endless lives in smart-matter panels orbiting this and other suns.

With time dilation the distances are not that vast. But even without, it is plausible to send biological matter in some form of hibernation across the galaxy in a space ship at ~0.3c. The energy requirements are pretty big, but you could use something like black hole star ships to efficiently convert matter into speed.