Writing from this time is often referred to as 'Early Modern English' in so far as the language has more in common with modern English than Old English, and is recognisable as the beginnings of how modern day English is spoken.

The first part of your question seems to address pronunciation, so in terms of how Shakespeare's English sounded, evidence indicates that Shakespeare's English sounded quite different from modern day English in its pronunciation. There are quite a number of examples of poetry where rhyming pairs form 'near misses' in modern English, but appear frequently enough that we can assume that they rhymed at the time they were written. Words that end in 'ity' like 'eternity' are put in rhyming pairs with words like 'lie', suggesting a pronunciation more like 'et-er-ni-tye' than modern 'et-er-ni-tee', for example. There are also puns in Shakespeare's work which no longer make sense, because they originally worked through a play on words involving a homophone and pronunciation of one of those words has diverged in modern pronunciation. A scholar called David Crystal has been working on evidence like this in order to establish what the Original Pronunciation of Shakespeare's works might have sounded like. You can see a video of actors working with OP here. It's also worth bearing in mind that then, as now, pronunciation would vary substantially across the country. Even if Crystal's Original Pronunciation comes very close to how London-based actors spoke, this wouldn't be representative of a speaker from a different region. Broadly speaking though, as far as we can tell, yes, Shakespeare's English did sound different, but not incomprehensibly so.

The other part of your question is more directed towards whether stage language would seem somewhat out of place when spoken off stage. In short, yes it probably would, but not really in the way that you describe by asking if words like 'forsooth' were commonly used. If someone went about their day speaking in the same way as people on stage spoke, it would probably seem about as odd as if someone spoke as though they had just wandered off the set of a modern day action movie. Just as the dramatic pauses, odd whisper, and punchy comebacks would seem odd (and truthfully, rather cringey!) in present day because it's a style of speech and use of language we associate with drama, so too would Shakespeare's stage writing sound a bit artificial. We can see this played out to some extent in the way that Shakespeare represents his comic characters from the lower classes. While Shakespeare certainly toys with this convention, typically, high status characters will speak in verse (usually iambic pentameter, either rhymed or un-rhymed) while characters marked as 'common' will depart from this formal register and speak in prose (see for example, Trinculo and co in The Tempest). The rarefied speech of the stage is deliberately contrasted with informal speech patterns that are more in line with everyday intonation.

In terms of embellishments - yes, Shakespeare's writing is often highly embellished, but not in the way that you are thinking. 'Forsooth', 'zounds', the use of 'thee/thy' are all pretty common in this period. They feel like embellishments to a modern reader as they have fallen out of common usage, but wouldn't have seemed odd in Elizabethan/Early Jacobean England (they map fairly well onto modern 'truly/honestly' and 'oh my god', and modern French 'tu'). Where Shakespeare's language is embellished in a way that would make it odd for every day conversation is in poetic language. When Prospero says 'We are such stuff as dreams are made on, and our little life is rounded with a sleep’ (The Tempest, act 4 scene 1) he is speaking in a register that would sound very odd if you were just going about your daily life, but which doesn't seem out of place on stage.

I hope this helps answer what you were looking for - let me know if you have any other questions!