You can change the tense without changing anything else about the narrative voice. The "I thinks" you mentioned have to do with the psychic distance of the narrator, not verb tense. As an example, I've rewritten your first two paragraphs into past tense:
Summers hadn't been summers in 6 years. It wasn't really cold, but every morning I woke up in Desdena, I wanted to dive deeper under the sheets. I knew the sun was shining, but I didn't really feel it -- it didn't burn my skin, it didn't boil my blood . . . it didn't drive me out of my bed so I could escape the heat by diving into the river. In Desdena, the sun shone, but nothing more than that.
I knew I had to get up, but the idea that I had to leave my lovely bed, to touch that floor with my bare feet? No thank you. I'd have given anything to be in my own room -- well, my other room -- again, and feel the promising heat of a bright day. But Threshral was a long way away. The unhappy thoughts drove me out of bed.
The floor was cold and the air felt cool on my warm skin, but it was bearable. I examined my dresses carefully. If I wanted to avoid going up North, I'd have to look overwhelming. Father wouldn't be convinced easily. I don't think he had ever been convinced, really, but, by the two moons, he wasn't going to send me up North! It freezes up there! I'd probably die, and the Swahlis would feast on my frozen corpse. How could he not see that?
The Threshralian gown wasn't really fit for the climate, but it always made me feel confident. I'd have worn it more, but the typical double split wasn't really fit for the prudish mentality of Desdena. But I did look glorious in it, and that's what I needed. I brushed my hair and braided it. If I couldn't convince Father, they'd cut it off. There was no way I was going -- they were not going to put me to sleep!
You'll note I changed very little, beyond the verb tense -- just relative locations and times, and did a little rephrasing of some of the commentary. The meaning is all the same, and I didn't credit any of the commentary at all. What you may notice, though, is that the past tense changes the feeling of the narrative. It feels choppier and reads faster than it did. That may be something you'll want to look at if you go with past tense.