Merhaba,

The dictionary I looked in defines "to sit down" as "oturmak", but this one means just "to sit" as well. So if I say "oturuyorum" that would mean I´m just sitting in place and not moving. So, is it possible to say "I´m sitting down" to emphasize that I´m moving down to my seat at the moment of speaking, but I´m not sitting yet?

Yep.

So then, how do you say that?

"Turkish: A Comprehensive Grammar" by Göksel and Kerslake says many verbs, like oturmak, can mean both entering into a state or being in a state. For those verbs, changing the aspect changes the meaning:

                       Tülay yanıma oturdu. (EVENT)

                       Tülay yanımda oturuyordu. (STATE)

The marking of progressive aspect here is YOR of course.

But there is a small group of verbs which in Turkish only denote entering into a state (EVENT): acıkmak ´to get hungry´, susamak ´to get thirsty´, yorulmak ´to get tired´, bıkmak ´to get bored´, kızmak ´to get angry´, kalmak ´be left´. So if you want to say you are in the state of thirst/boredom/anger you actually say you have entered that state using the past tense > susadım/bıktım/kızdım.

Once I was chatting with someone who was fasting. I wanted to say ´you must be thirsty´ but I used the wrong form and was misunderstood:

- Susuyorsun.

- Pardon biraz meşgulüm.

- ??

You say it same way. "Oturmak" can express a state or an action

Thank you everyone for help

Your example reminded me a conversation with my friend  

-Susar mısın?  (after he talked a lot) 

-Susamam,  yeni su içtim. 

- :S 

What is wrong with this conversation?

For me it sounds quite logical:

-Are you thirsty?

-I´m not thristy, I´ve just drunk water

When I say "Susar mısın",  I mean "will you please shut up"

In fact he responds like that on purpose because if I wanted to ask him if he was thirsty or not,  I would ask him like "Susadın mı".  I found this situation similar to abla´s because she says Susuyorsun and her friends understands that as "you are quiet"