Yes they met.
In chapter 31 Greatness of Ramesvara in Kotirudra Samhita of Shiva Purana (Pg No - 1381), Rama worshiped Shiva's earthen image to sought blessings of Lord Shiva. Shiva met Rama in that Island (now called as Rameswara) and blessed Him.
When Rama, the devotee of Shiva, took the water and about to drink, it
occurred to him at the will of Shiva. " I have not yet had sight of
Siva, the great lord, the bestower of all bliss. How can the water be
taken in? " After saying this, he drank water and then performed the
worship of Shiva's earthen image. He worshiped Shiva performing all
the sixteen ancillary services (Shodashopchar pooja) like invocation
etc. with devotion according to rules obtaining. After propitiating
Shiva with obeisances, divine eulogies etc. strenously, Rama joyously
prayed to Shiva with devotion. Satisfying with Rama's devotion, Shiva
blessed him to be victorious.
O Siva, if you are pleased, you shall stay here, O lord, to sanctify
the world and to render help to people. Suta said,
Thus implored Shiva took up the phallic image named Ramesvara. It is
now famous on surface of the world.
After defeating Ravana in War, Devas with Brahma and Shiva reached city of Lanka in Aerial Vehicle. (Verse 2)
Thereupon, Kubera the King of Yakshas, Yama the lord of death together
with the deceased ancestors, Indra the lord of celestials Varuna the
lord of waters, the illustrious Shiva the great deity who bears the
device of a bull as his banner and having three eyes, Brahma the
creator of all the worlds and the best among the knowers of sacred
knowledge all these together reaching the City of Lanka in aerial
cars, shining like the sun approached Rama.
Devas asked Rama why he ignored Seetha, when she was entering into the fire. Rama appealed them to describe in reality who he was. Brahma (the creator) proclaimed his divinity and eulogized him with hymns.
Lord Shiva then spoke to Rama.
"O lotus-eyed, long-armed, broad-chested, annihilator of enemies and
excellent among those upholding the cause of virtue! Thank heaven! You
accomplished this task. Fortunately has the fear born of Ravana -
which increased the severe darkness on the entire world has been
removed by you, on the battle-field, O Rama.You are fit to go to the
heaven, by comforting the depressed Bharata and the illustrious
Kausalya as well as seeing Kaikeyi and Sumitra, the mother of
Lakshmana having got the sovereignty of Ayodhya, bringing delight to
your friends, stabilizing the back-bone in the race of Ikshvaku,
getting excellent glory by performing a horse-sacrifice and by
granting riches to Brahmanas. This king Dasaratha, your father in this
mortal world and the highly glorious as well as the venerable person,
is seated in an aerial car, O Rama!. Having been delivered by you, his
(godly) son, the glorious king obtained the abode of Indra the lord of
celestials. you offer your respectful salutation to him, along with
Lakshmana, your brother".
Regarding the story of Sati meeting Rama, Sati and Shiva should have done Time Travel to meet Rama or they could have met another Rama. In this story, Shiva aka Rudra didn't meet Rama personally. Goddess Sati tests Rama of His divinity.
Sati actually belonged to Pitri Kalpa, previous Kalpa. Rudra waited for 21 Manvantaras to marry Shakti again. He married her (Parvati) in Vaivastara Manvantara of Sweta Varaha Kalpa. Ramayana happened in 24th Treta Yuga of Vaivastara Manvantara. We live in 28th Kaliyuga.