هذا القميص الرقيق مبلل. القميص الآخر جاف.

In Modern Standard Arabic, a demonstrative like هذا is normally followed by a definite noun, so هذا القميص is the standard way to say this shirt.

To an English speaker, it may look like this the-shirt, but in Arabic that is completely normal. The الـ on القميص does not make it sound odd or overly definite; it is just the regular pattern.

Because adjectives usually come after the noun in Arabic.

So:

• القميص الرقيق = the thin/delicate shirt
• literally: the-shirt the-thin

This is the normal adjective order in MSA, unlike English, where adjectives usually come before the noun.

When an adjective directly modifies a noun in Arabic, it normally agrees with that noun in:

• definiteness
• gender
• number
• case (in fully inflected MSA)

So because القميص is definite, the adjective must also be definite:

• القميص الرقيق = the delicate shirt

If the noun were indefinite, the adjective would also be indefinite:

• قميص رقيق = a delicate shirt

Because here they are not being used as adjectives inside the noun phrase. They are being used as the predicate of the sentence:

• هذا القميص الرقيق مبلل = This delicate shirt is wet
• القميص الآخر جاف = The other shirt is dry

In other words:

• الرقيق describes which shirt
• مبلل tells you what state it is in

That is why مبلل and جاف do not take الـ here.

In present-tense nominal sentences, Arabic usually does not use a separate word for is.

So:

• هذا القميص الرقيق مبلل literally looks like
• this the-shirt the-delicate wet

But the meaning is:

• This delicate shirt is wet

This is one of the most common differences between English and Arabic. The present-tense is/are is usually understood rather than spoken.

A very important clue is definiteness.

• القميص الرقيق مبلل = The delicate shirt is wet
• القميص الرقيق المبلل = The delicate wet shirt

So if مبلل had الـ, it would more naturally be read as another adjective modifying القميص. Without الـ, it works as the predicate: is wet.

This is a very useful reading strategy in Arabic:

• definite noun + definite adjective + indefinite adjective often means
  the [adjective] noun is [adjective]

Because قميص is a masculine singular noun, so it takes the masculine singular demonstrative هذا.

• هذا = this for masculine singular
• هذه = this for feminine singular

You can also see masculine singular agreement in the rest of the sentence:

• الرقيق
• مبلل
• الآخر
• جاف

All of these are masculine singular forms matching قميص.

الآخر means the other.

So:

• القميص الآخر = the other shirt

It is definite because it is describing a definite noun, القميص, and adjectives agree in definiteness.

Compare:

• القميص الآخر = the other shirt
• قميص آخر = another shirt / a different shirt

That difference is very important.

Because even words like الآخر behave like adjectives here, and Arabic adjectives normally come after the noun.

So the normal order is:

• القميص الآخر = the other shirt

Putting الآخر before the noun would not be the normal adjective order for this meaning in standard Arabic.

Yes. In fully vocalized MSA, the sentence could be written something like:

• هذا القميصُ الرقيقُ مُبَلَّلٌ. القميصُ الآخرُ جافٌّ.

A few useful points:

• Final case endings such as ـُ and ـٌ are often not written in ordinary text.
• In careful formal reading, they may be pronounced.
• In pause, they are often dropped in actual speech.
• الرقيق is pronounced ar-raqīq, not al-raqīq, because ر is a sun letter, so the l of الـ assimilates.

So learners often read the written form without all the final vowels, but it is still good to know that the full grammatical forms exist.