Here it says that he sent them on the ninth day, the 23rd. In Melachim I, 8:66 it says that he sent them on the eighth day, and there Rashi wrote that it was Tishrei 22!
'Rashi' #1, Malbim: On the eighth day they received permission to leave after the Chag 1 . On the next day, after the Chag, they all went together. Malbim (Melachim I, 8:66) - they could not leave on the night after Yom Tov, for Shemini Atzeres obligates Linah (spending the night in Yerushalayim).
'Rashi' #2: On the eighth day he gave to them permission to leave, and he sent those who live within Techum Shabbos, or anyone who wants to travel that day within the Techum. On the next day, he sent the entire nation.
Radak (citing Mo'ed Katan 9a): On the eighth day they received permission to leave. They did not travel that day, for it was Yom Tov. On the 23rd they received permission again, and went to their tents.
Malbim: He exempted them from the inauguration. This is why it says there only "Shilach", but not 'to their tents'!
Why were they "Semechim v'Tovei Lev"?
Shabbos 30a: They were "Semechim" because they enjoyed the radiance of the Shechinah. "V'Tovei Lev" - everyone's wife became pregnant and gave birth to a son.
Bereishis Rabah (35:3): A Bas Kol (voice from Heaven) announced that all of them are Bnei Olam ha'Ba.
What Tovah did Hashem do for David and Shlomo?
'Rashi' #1: He fulfilled His word that David's son would build the Beis ha'Mikdash.
'Rashi' #2: He fulfilled His word, that he chose Shlomo from all David's sons, for he was a Chacham.
What Tovah did Hashem do for Yisrael?
'Rashi' #1: It is like it says in Melachim I, 8:56 ? "Lo Nafal Davar?" (He fulfilled "v'Heni'ach Lachem mi'Kol Oyveichem" - Devarim 12:10.)
'Rashi' #2: They had a ruler, and dwelled with confidence and serenity ? "va'Yeshev Yehudah v'Yisrael" (Melachim I, 5:5). Radak (Melachim I, 8:66) - all kings served Shlomo and brought gifts to him.
Radak (Melachim I, 8:66, from Mo'ed Katan 9a): Yisrael were worried, lest they were liable for eating on Yom Kipur during the inauguration festivities. A Bas Kol announced that all of them are Bnei Olam ha'Ba.
