From Faro to Sagres is not, as far as I know, a camino or marked. I have done several long-ish walks on the coast over the years and there is definitely a path. But I don’t think you will find albergues. There are plenty of hotels, though, since you’ll be walking through the most heavily touristed beaches in Portugal.
Inspired by a travel report by Werner (Link), I have had the dream of a coastal hike in the Algarve for a long time: the core of this
www.wanderndeluxe.de
There is some information scattered around the forum about the Central route starting in Faro, but this takes you up the east side of Portugal, which isn’t what you asked about. Just in case, though, if you search for posts with “Faro” and you will find some good information on that route, like
this one and
this one by Nick, jungleboy.
And as you noted, there is a good amount of forum info about the Rota Vicentina. (for search purposes, note that there is no N before the C in Vicentina).
I personally think the western Alentejo coast is much prettier than the Algarve coast, but there are some very nice spots near Lagos and I think it would be a nice walk, lots of it on headlands.
And just posting in a link to a map that includes Faro and some options.
Ligando os pontos de culto e devoção ao Apóstolo Santiago no Sul de Portugal, particular foco no itinerário central que inicia em Faro - Loulé - Almodôvar - Castro Verde - Santiago do Cacém - Alcácer do Sal - Palmela - Lisboa - Santarém - Golegã - Tomar - Conímbriga - Coimbra - Santa Luzia -...
www.caminhosantiagoportuguessul.pt
I am by no means fully educated on these routes, and it is confusing.