In this post, I will compare two approaches to the Song of the Sea in Exodus 15: that of Frank Moore Cross in his classic Canaanite Myth and Hebrew Epic, and that of John Van Seters in Life of Moses. Cross dates the Psalm to Israel's pre-monarchic period (as in the twelfth-eleventh centuries B.C.E.), whereas Van Seters regards it as a post-exilic composition, which depends on J and P.
All I will say about the linguistic and grammatical arguments of Cross and Van Seters is that Cross believes that the Hebrew in Exodus 15 is early, whereas Van Seters argues that Exodus 15's linguistic and grammatical features "are also present in poetry of exilic and postexilic poems and prophecy" (Van Seters 147).
That said, I will start with a quote by Van Seters on pages 147-148 because doing so will crystallize the differences between him and Cross:
"Exodus 15 appears to be a highly eclectic poem with lines drawn from a variety of hymns of a quite general nature together with references to the sea event, the wilderness journey, and the eventual establishment of the cult on Mount Zion. It combines elements from both the Yahwistic and Priestly versions and is likely later than both. As [Brevard] Childs has pointed out, v. 8 seems to be a skillful artistic compilation of the two versions: 'By the wind of your nostrils the waters piled up, the floods stood up like a wall [ned], the deeps congealed in the heart of the sea.' Both the wind in J and the walls of water in P are reconciled in the imagery. This surely suggests direct literary dependence of the Song on both J and P."
According to Van Seters, the Song of the Sea in Exodus 15 appears to be aware of so many traditions in the Hebrew Bible---J and P's stories about the crossing of the Sea, the wilderness, Mount Zion, etc.---and so it obviously came about after those traditions were established. Because Van Seters dates J to the exile, it's not surprising that his date for the Song of the Sea is post-exilic.
But Cross takes the exact opposite approach. His point is that the Song of the Sea is early because it is unaware of traditions in the Hebrew Bible about the crossing of the Sea. According to Cross, Part I of the Song (Exodus 15:1b-12) does not present the Israelites crossing the parted (or wind-tossed) Sea on dry land, after which God withdraws his mighty wind and the waters come crashing down on the pursuing Egyptians. Rather, the Song just says that God threw Egyptians into the Sea. As Cross notes, there is a difference between J (whom he dates to the tenth century B.C.E.) and the Song of the Sea on the issue of the wind: In J, "The Egyptians are drowned when the wind ceases to blow and the sea returns to its perennial state", whereas, in the Song, "the divine wind overthrows Pharaoh and his host" (Cross 134).
I think what Cross wants us to picture when it comes to the Song of the Sea is God throwing Egyptians from the shore into the tumultuous Sea and drowning them with fierce waves (which would include the ned of v 8 that Van Seters mentions), rather than the Egyptians being in the middle of a parted Sea in pursuit of the Israelites, and dying after God closes the Sea back up. Cross says that other traditions about the Sea appear to be aware of the Song of the Sea, for they use some of its language, even if they misunderstand what the Song is saying. But the Song of the Sea was unaware of the other traditions, and, for Cross, that is because those other traditions did not exist yet: the Song is early.
But what about the Song's reference to God's sanctuary in v 17---as part of its story that God led the Israelites, enabled them to defeat the Canaanites, and planted them on the mountain of his inheritance, which God made for his sanctuary? Does that mean that the Song of the Sea mentions Zion, which would indicate that it is not pre-monarchical, since David and Solomon were the ones who established God's sanctuary on that particular mountain? Cross argues that the mountain of God's sanctuary in v 17 is the sanctuary of Gilgal, not Zion. But Gilgal is not on a mountain, right? Cross has an answer to that, on pages 142-143:
"One might complain that Gilgal was not on a high mountain and that its tent-shrine and twelve stelae were unprepossessing. Such matters were no problem to the ancient Canaanite or Israelite. A temple precinct in Sidon was called 'the high heavens'...A temple mound or platform constituted the counterpart of the cosmic mountain. It should be remembered also that Mount Zion itself was a low hillock overshadowed by the towering heights of the Mount of Olives; yet it was a mountain which 'at the end of days...shall be established as the top of the mountains/and shall be exalted above the hills.'...In short, the language of v 17 could apply to any Yahwistic sanctuary."
Cross believes that, at the Gilgal sanctuary, the Israelites celebrated the story of the crossing of the Jordan River, in light of which Cross interprets v 16, which talks about God's people passing by. For Cross, Exodus 15 was used in pre-monarchical worship at Gilgal. Eventually, what was believed to have happened at the Jordan was applied also to the Sea of Reeds, as we see in biblical poetry that pairs the two events (e.g., Psalm 114; Habakkuk 3:8). But, for Cross, that was not yet the case when the Song of the Sea came to be: it did not have a parting of the Sea, but it did imply that God led his people through the Jordan.