There are so many dramas about adults, I rarely watch school dramas. I really agree about the above mentioned Saigo Kara Nibanme no Koi, it's so funny and hearwarming. Here are a few more that I recommend, there are many more of course
Hirugao (romance: unhappily married women happen to meet people they fall in love with)
Ice World (detective/romance: a man meets a woman accused of murder and tries to find out the truth)
Itoshi kimi e (romance: female doctor meets photographer with a disease)
Long Goodbye (detective, great story)
Manhattan Love Story (comedy: people who come and go at a coffee shop where stuff just happens...)
Bara no nai hanaya (romance: single dad who owns a flower shop meets bilnd woman, secrets and lies)
Roosevelt Game (business/sport: the company is about to go down but the baseball club can save the day THIS DRAMA IS AWESOME)
Chase (crime/detective: it's an awesome hunter/hunted story about a criminal and the guy who tries to catch him)
Shinya shokudo (food/slice of life: a restaurant that opens at midnight, it's about the owner and the people who eat there, each episode has a different story and a different dish)
Date (romance/comedy: two adults who aren't like everybody else are, for different reasons, thinking about getting married)
Dinner (food/slice of life: an a very skilled chef tries to help a restaurant that isn't going so well to survive)
Underwear/aka Atelier (slice of life: a young woman comes to work for a famous underwear designer and learns about her goals in life)
Virgin Road (romance: pregnant woman finds a guy to act as her fiancé while she goes to visit her father)
Gunshi Kanbei (historical: the life of the brilliant strategist Kuroda Kanbei and the lives of those around him, AMAZING drama)