No native speaker of French and English. According to: https://en.wiktionary.org/wiki/a_drowning_man_will_clutch_at_a_straw

"a drowning man will clutch at a straw"

has an equivalent shorter version

"grasp at straws".

According to https://www.linguee.fr/anglais-francais/traduction/grasping+at+straws.html one gets some ways to convey the desired meaning in French (not necessarily the Metropolitan French) such as:

It is total grasping at straws. On fait flèche de tout bois.

Or are we clutching at straws here? Ou parlons-nous à tort et à travers ?

and so on.

According to https://dictionnaire.reverso.net/anglais-francais/grasping+at+straws one may also use:

se raccrocher aux branches

e.g.

I mean, they must be really grasping at straws. Ils doivent vraiment se raccrocher aux branches.

Cf.

https://fr.wiktionary.org/wiki/se_raccrocher_aux_branches

https://www.languefrancaise.net/Bob/35991

EDIT

(Thanks to native speaker @Pas un clue).

‘Se raccrocher aux branches’

is an option. In the exact contexte, one may use

«on tente de se raccrocher à n’importe quelle branche»;

or

«…à n’importe quoi»;

but it’s not as colorful as the original English idiom. A couple of examples of usage in French: 1; 2. A literal translation from English to French is provided here. The image may not be customary in French, but very understandable nonetheless.

EDIT 2

(Thanks to native speaker @Personne)

Another way to convey the same meaning:

Se donner au premier saint venu.