Τα χείλη μου είναι πολύ ξηρά σήμερα, γιατί δεν πίνω αρκετό νερό.

In Greek, it is very common to use the definite article with body parts and with possessive words like μου.

So:

• τα χείλη μου = literally the lips my
• natural English translation = my lips

This is the normal Greek way to say it.
Using χείλη μου without the article is much less standard here.

Χείλη is plural and means lips.

The singular is:

• το χείλος = lip

So:

• το χείλος = the lip
• τα χείλη = the lips

In this sentence, Greek uses the plural because it is talking about your lips in general.

Greek possessive forms like μου, σου, του, της usually come after the noun.

So Greek says:

• τα χείλη μου = my lips
• το βιβλίο μου = my book
• η μητέρα μου = my mother

This is one of the basic word-order differences from English.

Because adjectives in Greek must agree with the noun in gender, number, and case.

Here:

• χείλη is neuter plural
• so the adjective must also be neuter plural

That gives:

• ξηρά = dry, agreeing with χείλη

A useful pattern is:

• ξηρός = masculine singular
• ξηρή = feminine singular
• ξηρό = neuter singular
• ξηρά = neuter plural

So τα χείλη μου είναι πολύ ξηρά means my lips are very dry.

Here πολύ is being used as an adverb, meaning very, so it stays πολύ.

It modifies the adjective ξηρά:

• πολύ ξηρά = very dry

Compare that with πολύς / πολλή / πολύ, which are adjective forms meaning much/many and do change depending on the noun.

Examples:

• πολύ νερό = a lot of / much water
• πολλοί άνθρωποι = many people
• πολλές μέρες = many days

But in your sentence:

• πολύ ξηρά = very dry

So here πολύ does not agree with χείλη.

Because Greek often leaves out subject pronouns when they are already clear from the verb ending.

• πίνω already means I drink
• δεν πίνω = I do not drink

The ending -ω tells you it is first person singular.

You can add εγώ if you want emphasis:

• εγώ δεν πίνω αρκετό νερό = I don’t drink enough water

But without emphasis, Greek normally just says δεν πίνω.

Yes. Γιατί can mean both:

• why
• because

The context tells you which one it is.

Here it comes after a statement and introduces a reason:

• ..., γιατί δεν πίνω αρκετό νερό.
• ..., because I don’t drink enough water.

If it were a question, it could mean why?

For example:

• Γιατί είναι ξηρά τα χείλη σου; = Why are your lips dry?

Greek also has επειδή for because, and that would work here too, but γιατί is very common and natural.

Αρκετό means enough or sufficient.

So:

• δεν πίνω αρκετό νερό = I don’t drink enough water

It does not mean exactly the same as a lot of water.

Compare:

• πίνω πολύ νερό = I drink a lot of water
• πίνω αρκετό νερό = I drink enough water

In your sentence, the idea is that the speaker is not drinking a sufficient amount.

Because νερό is being used as an uncountable noun, like water in English.

So Greek naturally says:

• πίνω νερό = I drink water
• πίνω αρκετό νερό = I drink enough water

You usually do not use the article here unless you mean some specific water already known in the context.

It is the same form for both is and are.

So:

• είναι can mean is
• είναι can also mean are

You understand it from the subject.

Here the subject is:

• τα χείλη μου = my lips

So είναι means are in this sentence:

• Τα χείλη μου είναι πολύ ξηρά = My lips are very dry

Yes, Greek word order is fairly flexible, especially with adverbs like σήμερα.

The sentence has:

• Τα χείλη μου είναι πολύ ξηρά σήμερα

This is perfectly natural.

But Greek could also say:

• Σήμερα τα χείλη μου είναι πολύ ξηρά
• Τα χείλη μου σήμερα είναι πολύ ξηρά

The meaning stays basically the same, though the emphasis can shift a little.
The version you have sounds natural and smooth.