Two things that readily come to mind:

• The whole To The Moon series hits really hard. Note that if it sounds interesting to you to see the perspective of doctors meddling with people's memories before they die (so they die on a happier memory, basically), don't watch anything major about this. Giving too much away ruins the impact.
• If you are fine with the low-ish production values, the old-ish VN Katawa Shoujo works really well. Instead of showing ordinary people in extraordinary situations, it shows extraordinary people in completely ordinary situations. But as a result of how our memories of our teenage years distort with age, this works really well on adults. In particular for me, Rin's route was really good and made me cry a lot.