وضبت الاغراض بالكرتونة وسكرتها بالشريط اللاصق.

In Levantine, وضّب / يوضّب often means to arrange, put in order, pack up, get things together.
So وضبت الاغراض means something like I packed the things / I put the stuff together.

In this sentence, packed is the most natural translation.

In everyday Arabic writing, especially dialect writing, people often leave out the shadda (the doubling mark). So:

• وضبت = casual spelling
• وضّبت = more explicit spelling

They represent the same word here. The verb is pronounced with a doubled ḍ sound.

الأغراض / الاغراض means the things, the stuff, the belongings, the items.

In spoken Levantine, غراض / أغراض is very common for everyday stuff.
So وضبت الاغراض is basically I packed the stuff.

In Standard Arabic, you would usually write الأغراض.
In informal dialect writing, people often drop the hamza in spelling, so الاغراض is very common online and in texting.

So this is mostly a spelling habit of casual writing, not a different word.

بالكرتونة is:

• بـ = in / into / with, depending on context
• الكرتونة = the box / the carton / the cardboard box

Here it means in the box or into the box.

So:

• وضبت الاغراض بالكرتونة
  = I packed the stuff in/into the box

In Levantine, بـ is extremely flexible and often covers meanings that English splits into several prepositions.

Usually كرتونة means a carton or cardboard box.

So it is more specific than a general box.
A more general word for box is صندوق.

So الكرتونة suggests something like a packing box, shipping box, or cardboard container.

Because the verb already shows the subject.

• وضبت = I packed
• سكرت = I closed / I sealed

Arabic often does not need a separate subject pronoun when the verb already makes it clear.
You could add أنا for emphasis, but it is not necessary.

The -ها means it / her as an attached object pronoun.

Here it refers back to الكرتونة, which is a feminine noun in Arabic. So:

• سكرت = I closed
• سكرتها = I closed it

Because كرتونة is feminine, the pronoun is -ها.

If the object were masculine, you would use -ه instead.

Both are possible, depending on context.

The verb سكّر in Levantine means to close. But when you say:

• سكرتها بالشريط اللاصق

the context makes it sound like I sealed it with tape, not just I shut it.

So the literal idea is I closed it, but the natural English meaning here is often I sealed it.

It means with the adhesive tape or simply with tape.

Breakdown:

• بـ = with / using
• الشريط = the tape / the strip
• اللاصق = the adhesive / sticky

So together:

• الشريط اللاصق = adhesive tape
• بالشريط اللاصق = with adhesive tape

Because بـ is a very broad preposition in Arabic, especially in spoken dialects.

In this sentence:

• بالكرتونة = in/into the box
• بالشريط اللاصق = with the tape

This is completely normal. Arabic often uses one preposition where English uses several different ones.

This is because of sun-letter assimilation.

The Arabic definite article is written الـ, but its l sound changes before certain letters.

• الشريط starts with ش, a sun letter
  so ال + شريط is pronounced ash-sharīṭ / ish-sharīṭ in connected speech

• اللاصق starts with ل, which also causes assimilation
  so it is pronounced with a doubled l sound: allāṣeq / illāṣeq

In Levantine connected speech, بالشريط اللاصق often sounds like:

• bish-shrīṭ illāṣeq

A common Levantine-style pronunciation would be something like:

waḍḍabet l-aġrāḍ bil-kartūne w sakkartha bish-shrīṭ illāṣeʔ

A few notes:

• the short vowels are not written in normal Arabic spelling
• ق in لاصق is often pronounced as a glottal stop in many urban Levantine accents, so the end may sound like -eʔ
• pronunciation varies by country and city

Yes, it sounds natural.

A speaker might also say similar versions, such as:

• حطيت الاغراض بالكرتونة وسكرتها بالشريط اللاصق
• لمّيت الاغراض بالكرتونة وسكرتها
• وضبت الغراض بالكرتونة ولزّقتها بالشريط

These are just small variations in wording. The original sentence is perfectly understandable and natural.