Written by the team of Rabbanim from the Orthodox Union:

Rabbi Eli Gersten , Rabbi Yaakov Luban and Rabbi Moshe Zywica

Yes, the requirement to eat three meals on Shabbat applies even when Rosh Hashanah falls on Shabbat. Although davening ends late, we somehow have to eat two meals on Shabbat day.

In the evening (motzei Shabbat) will be the second day of Yom Tov, and according to halacha  one should not begin a meal three hours before Yom Tov (so that the evening Yom Tov meal will be consumed with an appetite). It may be difficult to eat two meals between the end of Musaf and  three hours before sunset. Nonetheless, the Mishnah Berurah (529:8) writes that if one did not eat Shalosh Seudot before the cutoff time, it should be eaten until sunset  because the mitzvah of Shalosh Seudot outweighs the honor of Yom Tov.

To avoid this problem and fit everything into the very limited  time frame, some split the Shabbat meal in half by bentching, then taking a small break, and after that reciting hamotzi once again before the three-hour cut-off point.  

Shalosh Seudot Times to Be Finished By:

Brooklyn–4:38
Lakewood–4:39
Passaic–4:39
Manhattan–4:38
Queens–4:38
Monsey–4:39
Nassau–4:37
Baltimore–4:48
Miami–4:54
Detroit–5:16

Cleveland–5:10
Chicago–4:34
Los Angeles–4:31
Phoenix–4:06

(Source: myzmanim.com)