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Three things are critical to see:

1. Ahaz's refusal to do what the LORD asked of him is wearying to God, as much of the behavior of the lineage of David had been (both the "you" references in the verse are plural, not singular, in the Hebrew). So Ahaz is being addressed, but obliquely in his connection to the lineage of David, which leads to the second point...
2. Isaiah says "my God" not "your God" or "our God".
3. Ahaz is no longer the sole referent, but rather the referent expands to the "house of David" (any and all those in the line of David, Ahaz included, but not exclusively or even necessarily to him alone). So the prophecy is for the "house of David" to pay attention to, such that when it is fulfilled, they will take notice of it.

Two things are critical to see:

1. Ahaz's refusal to do what the LORD asked of him is wearying to God, as much of the behavior of the lineage of David had been (both the "you" references in the verse are plural, not singular, in the Hebrew). So Ahaz is being addressed, but obliquely in his connection to the lineage of David, which leads to the second point...
2. Ahaz is no longer the sole referent, but rather the referent expands to the "house of David" (any and all those in the line of David, Ahaz included, but not exclusively or even necessarily to him alone). So the prophecy is for the "house of David" to pay attention to, such that when it is fulfilled, they will take notice of it.

Two things are critical to see:

1. Ahaz's refusal to do what the LORD asked of him is wearying to God, as much of the behavior of the lineage of David had been (both the "you" references in the verse are plural, not singular, in the Hebrew). So Ahaz is being addressed, but obliquely in his connection to the lineage of David, which leads to the second point...
2. Ahaz is no longer the sole referent, but rather the referent expands to the "house of David" (any and all those in the line of David, Ahaz included, but not exclusively or even necessarily to him alone). So the prophecy is for the "house of David" to pay attention to, such that when it is fulfilled, they will take notice of it.

Note that the only immediate point of v.15-16 is that one land abhorred by Ahab that has two kings over it will cease to have kings by the time this child of Isa 7:14 is very old. Almost unanimously commentators will identify these two kings with Rezin and Pekah (Remaliah's son), identifying the single land as symbolically illustrating the united forces of two lands (quotes from commentaries found at the preceding link):

But this is God's message, and though the sign is broadly to the house of David, the last part of v.16 relates it to Ahab with "The land that thou abhorrest shall be forsaken of both her kings" (KJV). Most later translations have "dread" in place of the idea of abhor, as the word can have either idea (see BDB), and Ahaz's fear is known from v.4.

Three things are critical to see:

1. Ahaz's refusal to do what the LORD asked of him is wearying to God, as much of the behavior of the lineage of David had been (both the "you" references in the verse are plural, not singular, in the Hebrew). So Ahaz is being addressed, but obliquely in his connection to the lineage of David, which leads to the second point...
2. Isaiah says "my God" not "your God" or "our God".
3. Ahaz is no longer the sole referent, but rather the referent expands to the "house of David" (any and all those in the line of David, Ahaz included, but not exclusively or even necessarily to him alone). So the prophecy is for the "house of David" to pay attention to, such that when it is fulfilled, they will take notice of it.

Note that the only immediate point of v.15-16 is that one land abhorred by Ahaz that has two kings over it will cease to have kings by the time this child of Isa 7:14 is very old. Almost unanimously commentators will identify these two kings with Rezin and Pekah (Remaliah's son), identifying the single land as symbolically illustrating the united forces of two lands (quotes from commentaries found at the preceding link):

But this is God's message, and though the sign is broadly to the house of David, the last part of v.16 relates it to Ahaz with "The land that thou abhorrest shall be forsaken of both her kings" (KJV). Most later translations have "dread" in place of the idea of abhor, as the word can have either idea (see BDB), and Ahaz's fear is known from v.4.

As I sketched out in this answer, Ahaz's refusal to ask for a sign is critical to interpreting what follows. For what follows is not best conceived in context as a sign "designed to signify to Ahaz that the land will be delivered from its calamities." Rather, it is indeed a prophecy of judgment.

1. A child will be born under specific circumstances (see the previous answer linked to with respect to the virgin aspects, but also below for a key footnote I had there that is really more related to this question [hence why it was a footnote in the other question]), a child whose name implies God is with His people again (Isa 7:14-16)
2. But more immediately, judgment to Judah will be coming (i.e., God not with His people). That judgment is not by the Syria/Ephraim alliance (that was already made clear), but rather by Assyria (Isa 7:17-25), the nation Ahaz would look to for help against the alliance (2 Kings 16), which judgment comes about in Ahaz's son Hezekiah's days (2 Kings 18), though Jerusalem is spared (and Judah for a time) because of Hezekiah's looking/prayer to God (2 Kings 19:20).

As I sketched out in this answer, Ahaz's refusal to ask for a sign is critical to interpreting what follows. For what follows is not best conceived in context as a sign "designed to signify to Ahaz that the land will be delivered from its calamities." Rather, it is indeed a prophecy of judgment.

1. A child will be born under specific circumstances (see the previous answer linked to with respect to the virgin aspects, but also below for a key footnote I had there that is really more related to this question [hence why it was a footnote in the other question]), a child whose name implies God is with His people again (Isa 7:14-16)
2. But more immediately, judgment to Judah will be coming (i.e., God not with His people). That judgment is not by the Syria/Ephraim alliance (that was already made clear), but rather by Assyria (Isa 7:17-25), the nation Ahaz would look to for help against the alliance (2 Kings 16), which judgment comes about in Ahaz's son Hezekiah's days (2 Kings 18), though Jerusalem is spared (and Judah for a time) because of Hezekiah's looking/prayer to God (2 Kings 19:20).