As of Minecraft version 22w12a, there is a new field warden_spawn_tracker in the player's NBT data.
Like almost all other such fields, there should be entity API methods to query and modify the various values in that set.

Example: Create a Scarpet app that implements the /warden_spawn_tracker command from 1.19 Deep Dark Experimental Snapshot 1.

The following is to be added to the documentation (entities.md) when this is implemented. Everything below is subject to change.

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query(e, 'warden_warning_level')

The number of times the player has activated a Sculk Shrieker (0 to 3 inclusive), or null if e is not a player.
If this is 3, the next Sculk Shrieker activation will spawn a Warden.

query(e, 'warden_warning_cooldown')

The number of ticks before the player can activate another Sculk Shrieker, or null if e is not a player.

query(e, 'warden_warning_ticks')

The number of ticks since the player last activated a Sculk Shrieker, or null if e is not a player.
At 12000, it resets to zero and warden_warning_level described above is decremented.

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modify(e, 'warden_warning_level')

Set the number of times the player has activated a Sculk Shrieker (0 to 3 inclusive). Setting this to 3 causes the next activation within 12000-warden_warning_ticks ticks to spawn a Warden. Has no effect if e is not a player.

modify(e, 'warden_warning_cooldown')

Set the number of ticks before the player can activate another Sculk Shrieker. Has no effect if e is not a player.

modify(e, 'warden_warning_ticks')

Set the number of ticks since the player last activated a Sculk Shrieker.
Setting this to 12000 or above resets it to 0 and decrements warden_warning_level. Has no effect if e is not a player.