ומאילם נסעו לאלוש, שנאמר ויסעו מאילם ויבאו כל עדת בני ישראל אל מדבר סין (שמות טז א) (הוא אלוש) בחמשה עשר יום לחדש השני לצאתם מארץ מצרים (שם), ואחד בשבת היה, הא למדנו שראש חדש אייר באחד בשבת היה, ועוד למדנו שהיו ישראל אוכלין מעוגה שהוציאו בידם ממצרים כל שלשים יום, ובו ביום כלתה, ולערב אכלו את השליו ולמשכים לקטו את המן, ובאלוש נתנה להם השבת, ושם עשו שבת ראשונה, שנאמר וישבתו העם ביום השבעי (שם טז ל), באחד בשבת בכ"ג באייר נסעו מאלוש ובאו להם לרפידים
From Eilim they traveled to Alush, as it says, "And they traveled from Eilim and the entire congregation of Bnei Yisrael came to Midbar Sin on the 15th of the second month from leaving Mitzraim" (Shemos 16:1). It was a Sunday; this teaches us that Rosh Chodesh Iyar was a Sunday. This also teaches us that they ate from the cakes which they brought in their hands from Mitzraim for thirty days, and on that day, it ran out. That evening, they ate the slav, and when they woke up, they collected the mann. In Alush they received Shabbos, and there they made their first Shabbos, as it says, "And the nation rested on the seventh day" (ibid. v. 30). On Sunday, 23 Iyar, they travelled from Alush and came to Refidim.
This seems to be self-contradictory: first it insists that 15 Iyar was a Sunday, then it says that the following Sunday was the 23 Iyar! Further, the Braisa proceeds to explain that the Torah was given on 6 Sivan on Shabbos; this fits with the 23 Iyar being a Sunday, but not with 15 Iyar being a Sunday.
Further, the Braisa insists that 15 Nissan is a Friday. This only works if 15 Iyar is a Sunday, assuming Nissan is 30 days, or Shabbos, if Nissan is 29 days.
Were the Braisa to say that 15 Iyar is a Sunday, 22 Iyar is a Sunday, and 29 Iyar is a Sunday, I'd have no problem; Seder Olam Rabbah agrees with the Chachamim in Shabbos 87b that the Torah was given on Shabbos, 6 Sivan, but argues on the Braisa there that Bnei Yisrael left Mitzraim on a Thursday. However, it does not say this.
Is this simply a typo in Seder Olam that crept in over the generations? Or am I missing something here?