You would say:

Je suis dans la merde jusqu’au cou (= I’m in shit to the neck)

In addition to describing/declaring the state of being in a bad situation by coming right out and using the arguably vulgar image of one actually being in deep s–t/doo-doo/caca (as suggested in your question and as confirmed in the good accepted answer), you could also consider using the figure of style/speech known as l’antiphrase to convey this message ironically with:

On [n’]est pas dans la
merde ! …

which, according to the first two posts on the linked WordReference.com thread discussing that expression, means “things are going wrong” and “we are in big shit!”, respectively.

For an example of this usage, here’s a comedy sketch by Les Inconnus (via Dailymotion.com) where “[Avec tout ça,] On [n’]est pas dans la merde” is used repeatedly to clearly (in my opinion) mean the contrary, i.e., that “l’on est [vraiment] dans la merde.”

(Please note that this ironic expression could also be seen as an understatement, therefore perhaps qualifying it as an example of la litote, but I see it more as a prime example of l’antiphrase.)

The above links for antiphrase and litote and their definitions below are from the website of the Office québécoise de la langue française:

antiphrase n. f. [Corresponding] Terme anglais: antiphrasis
Définition:
Figure de style qui consiste à employer un mot, un syntagme ou une phrase dans un sens contraire à son sens véritable.

litote n. f. [Corresponding] Terme anglais: litotes [& English] Synonymes: meiosis/understatement
Définition:
Figure de style qui consiste à employer les mots de façon atténuée pour suggérer davantage que ce qui est dit.

Finally, as a slight (and unnecessary, imo) caution about whether this ironic version is vulgar enough for your purposes, since a literal reading of this antiphrase/litote actually conveys the absence of the [arguably] vulgar word (and image) at issue, I suppose that it could technically be considered less vulgar than the directly-stated, “non-antiphrasic” version.
However, since the word at issue is still present and, more importantly, since the [arguably vulgar] figurative meanings (and images) of both versions are essentially the same, I don’t think that the actual level of vulgarity that you seek would be reduced by using the ironic version.

Je suis bien dans la merde (I’m really in shit)

Je suis dans la merde bien profond (I’m in deep shit)

Je suis dans la merde jusqu’au cou (I’m in shit up to my neck)

Je suis dans une putain de merde (I’m in some fucking shit)

Je suis niqué (I’m fucked)