I think you overestimate the complexity of medieval cryptography -- a Vigenère cipher, for instance, was still rather "high-grade" at the time yet for someone well-versed in its use would require nothing more than a single piece of parchment, if even that much1 -- but no matter. We'll run with your list of requirements, because there's a very interesting answer: steganography.
In our modern digital age, you may think of steganography as the black art of hiding a secret message within the "noise" of a digital photo, for instance, but it's far older than that. One of the oldest forms of steganography, and perhaps most useful to your requirements, is using a predetermined method of encoding to embed a secret message within a paragraph (or more) of innocuous text. For instance, your message might be comprised of every 10th word, while the rest is literally just filler intended to mask its very presence. More complex examples rely on using a cover with windows cut into it in random places, concealing the innocuous text and revealing only the text relevant to the secret message. This gives you the ability to send your secret messages without even revealing that you're sending a secret message at all, giving you absolute deniability should you be caught. You could even employ an encoding scheme such that rather than individual words, you hide particular letters (3rd letter of each 5th word), which could themselves form a Vigenere cipher of your actual message. While quite difficult, this could even be used in spoken language, though the encoding would necessarily have to be relatively simplistic to be able to be deciphered that quickly.
This doesn't (necessarily) give you the appearance of being a new language, however. (Not unless your innocuous text is itself something else.) For that, what you want is either an obscure, real language (see e.g. code-talkers) that your secret society could use for their purposes, or else they need to devise their own constructed language (which is often offered as one possible explanation for the mysterious Voynich manuscript, for instance). You could even go one better and encode the written form in its own alphabet, for that extra little bit of mysteriousness, though someone who can see and hear both forms of the same message would likely be able to quickly decipher the alphabet (unless you deliberate obfuscate the sounds somehow).
This is really your best bet for meeting all your requirements: A constructed or obscure language used solely for the purpose of communicating esoterically with other members of your secret society. Most likely you wouldn't need to devise as rich a language as e.g. English, just come up with the syntax and vocabulary for the most basic forms of communication, and revert to another common tongue in the few instances where it actually becomes necessary.
There is one more very interesting option: A substitution cipher specifically designed to encode a well-known language (e.g. English) into something that merely looks like another language, crafted in such a way as to remain readable. A lot of online "language generators" employ a scheme like this, and it's frankly amazing how easily the results could actually be read aloud if one wanted to. At its most basic, this involves enciphering one vowel as another, and one consonant as another; it could be made more complex by using a different key (a la Vigenere) on a per-word basis (rather than per-letter as in Vigenere), though this risks destroying the ability to effectively speak it.
1 I used to be super gung-ho into secret messages, and -- albeit with effort -- I could read even complex substitution ciphers like Vigenere (though I didn't know the term at the time) almost as if they were their plain-text equivalents. This required, however, being not only familiar with the method, but having used the same key repeatedly -- which may in fact be the case for a secret society, even though each re-use of the same dramatically increases the likelihood that it will be deciphered by your enemies.