Πριν φύγω, θέλω να βάλω τα χαρτιά μου στη συρταριέρα και να ισιώσω λίγο την κορνίζα.

Because Greek uses φύγω here to refer to a single completed action in the future: before I leave.

• φεύγω = present/imperfective, more like I leave / I am leaving
• φύγω = aorist subjunctive form, used here after πριν for a one-time event

So:

• Πριν φύγω = before I leave
• not before I am leaving

This is a very common pattern in Greek.

Φύγω is the aorist subjunctive form of φεύγω (to leave).

In this sentence, it appears after πριν, but it is the same basic form you also see after να or θα in related structures:

• να φύγω = for me to leave / that I leave
• θα φύγω = I will leave
• πριν φύγω = before I leave

So even though English learners often first meet φεύγω, Greek very often switches to φύγω when talking about one complete act of leaving.

Because the speaker wants to talk about doing each action once, as a complete action.

• να βάλω = to put once / put away
• να ισιώσω = to straighten once

If you used the present/imperfective forms:

• να βάζω
• να ισιώνω

that would suggest something more ongoing, repeated, or habitual.

Here the meaning is clearly:

• put the papers away
• straighten the frame

So the aorist subjunctive forms are the natural choice.

Θέλω να means I want to.

It is made of:

• θέλω = I want
• να = a particle that introduces the following verb in the subjunctive

So:

• θέλω να βάλω = I want to put
• θέλω να ισιώσω = I want to straighten

This is one of the most common Greek patterns:

• θέλω να πάω = I want to go
• θέλω να δω = I want to see
• θέλω να φύγω = I want to leave

In standard Greek, you normally repeat να before the second verb as well:

• θέλω να βάλω ... και να ισιώσω ...

That is because both verbs belong to the same subjunctive structure.

So Greek prefers:

• να βάλω
• και να ισιώσω

rather than leaving the second να out.

For learners, the safest rule is: when two subjunctive verbs are linked with και, repeat να.

Στη is the contracted form of σε τη(ν).

Here:

• σε = in / into / to
• τη(ν) = the (feminine accusative)

So:

• στη συρταριέρα = in / into the chest of drawers

This contraction is extremely common:

• σε + τη(ν) = στη
• σε + το = στο
• σε + τα = στα
• σε + τον = στον

Because the preposition σε takes the accusative case in Modern Greek.

So in:

• στη συρταριέρα

the noun is in the accusative, even though in English we just say in the chest of drawers without thinking about case.

The same happens in many everyday phrases:

• στο σπίτι = in the house / to the house
• στην πόλη = in the city / to the city
• στο τραπέζι = on the table

Because the normal Greek way to say my papers is:

• τα χαρτιά μου

Greek usually places the weak possessive form (μου, σου, του, etc.) after the noun phrase:

• το βιβλίο μου = my book
• τα κλειδιά μου = my keys
• τα χαρτιά μου = my papers

Putting μου before the noun would not be the normal neutral pattern here.

Greek uses the definite article much more often than English.

So:

• τα χαρτιά μου literally looks like the papers my
• την κορνίζα = the frame

This is normal Greek, especially when talking about:

• specific objects
• possessed things
• direct objects that are already known or identifiable

So even when English might say my papers or straighten the frame, Greek naturally keeps the article.

They are both in the accusative case because they are direct objects of the verbs.

• να βάλω τα χαρτιά μου
  The thing being put is τα χαρτιά μου
• να ισιώσω λίγο την κορνίζα
  The thing being straightened is την κορνίζα

So if you ask what? after the verb, the answer is usually the direct object, which in Greek is typically accusative.

Λίγο means a little, a bit, or slightly.

So:

• να ισιώσω λίγο την κορνίζα = to straighten the frame a little / a bit

Here it probably suggests a small adjustment rather than a major action.

In everyday Greek, λίγο is very common and often softens what you say:

• Περίμενε λίγο. = Wait a moment.
• Άνοιξε λίγο το παράθυρο. = Open the window a little.

It can cover both ideas, depending on context.

Greek σε / στη often corresponds to English:

• in
• into
• sometimes on / onto, depending on the noun and situation

Here the natural English translation is something like:

• put my papers in the chest of drawers
• put my papers into the chest of drawers

The Greek phrase itself does not need a separate word to force the distinction the way English sometimes does.

Yes, Greek word order is fairly flexible, though the given sentence is the most neutral and natural order.

The original sentence:

• Πριν φύγω, θέλω να βάλω τα χαρτιά μου στη συρταριέρα και να ισιώσω λίγο την κορνίζα.

sounds smooth and neutral.

You could move things around for emphasis, for example:

• Πριν φύγω, θέλω να βάλω στη συρταριέρα τα χαρτιά μου...

That might put a little more focus on where the papers are going.

But for learners, the original word order is a very good model to follow.

Greek usually prefers a full finite clause here:

• Πριν φύγω = Before I leave

English often likes before leaving, but Greek commonly uses the verb with an expressed subject form built into it.

So although English can use a gerund, Greek naturally says:

• Πριν φύγω
• literally, Before I leave

This is one reason Greek sentences may feel more verb-based than their English equivalents.