Eight years ago, in 2015, I wrote an epic rant about the use of language in fantasy. Unfortunately it was buried in a newsletter about game development, so after a while I couldn't remember where that was anymore. But what is buried always comes out again sooner or later. Here it is, still as relevant.
Let's start with faux-Shakespearean English. TVTropes has an entire article about it, but the tl;dr version is, peppering your characters' speech at random with old verbal forms misremembered from King James' Bible does not count as "medieval flavor". For one thing, the Middle Ages officially ended over two centuries before Old Will's time. (Now, if you're going for a quasi-Renaissance setting, that's different, but how many fantasy writers do that?) Second, you most likely don't know the rules of early 17th-century English, let alone older dialects, and you're making a big ridiculous mess of it.
So what is there to do? One good idea is to do nothing in particular. Just like Captain Picard speaks modern English (because most people can't begin to guess what we'll talk like in a few centuries), your medieval characters can stick to the language of their audience. Of course, you'll want to avoid ultra-modern words such as psychology, but for the most part you should be good. Another would be to read older books — but not too old; 19th century should do fine — and see how writers used to word things back then, because it's more than just a matter of vocabulary. You want to pick just enough mannerisms from times past that your readers might feel the fingers of days long gone clinging to the edge of your utterances. Just don't overdo it, because readers will mock you.
The other issue is imposing on your readers the boring cosmology of yet another Standard Fantasy Setting. How many different ways can you tell people that blah blah orcs, blah elves and dwarves, blah dragons? Don't get me wrong, exposition can be great. It's a tool in the writer's arsenal, and anyone who tells you to avoid it is a fraud. But exposition should give the reader useful information. What is unique about your setting? What does the reader need to know right now that can't be shown through a bit of action down the road?
Speaking of that, the Standard Fantasy Setting is a useful trope. It's the perfect shortcut — the reader will instantly figure out the basic rules, and you can get right on with the story. But that's yet another argument for avoiding lengthy introductions that say nothing new. On the flipside, all the weight now rests on the characters. If they fail to capture the reader's interest and sympathy, there's no sense of wonder to fall back on. Are you a good enough writer?
That is the question (still).