Indeed, indigent implies extreme poverty.
Social services can issue what is called a "certificat d'indigence" (certificate of extreme poverty). Its issuance is required for instance when a deceased person has neither relatives nor money to pay for its funerals. The city will fund the burial of persons holding such a certificate.
In jails, "indigents" are the people with no support from the outside. They have to do their own laundry in the shower room (usually detainees give their laundry to their visiting family).
"Démuni" (which I confirm is almost always used as an adjective) basicaly means that you are lacking something in a great way.
2 uses come to my mind :
"Se retrouver démuni face à" : to face [something] powerless / helpless
Politics will often refer to "les plus démunis d'entre nous" : those among us who have the least [literaly the most resourceless among us]