You are over thinking this.
You know where you need to live. You need to be there by February. OMG that is so soon! Just based upon that, you need to get your house on the market now.
If it sells fast (Yay!), then you will still have at least a few weeks to figure out where you are going to live. I've never seen a house sale close in less than two weeks, for example. Most take 1-2 months to close. You can live there that whole time. You can try to negotiate a rent-back later if you need to, but I wouldn't even mention that to the agent right now. You don't know what kind of offers you will get until you start getting offers, and you can't get offers until prospective buyers are looking at it. Get it on the market! Get some bids!
When I needed to move fast, I sold my home to OpenDoor. I have a large dog and didn't want to deal with the hassle of taking her out for showings. Plus her presence meant pet odors, which buyers don't like, and I'd have to vacuum up her fur pretty much daily, which I don't like. Not sure if they buy homes in your area, but they were really easy to work with. I got an offer from them right away, but they give you a long time to think about their offer, so I took that time to have other similar companies give me offers (Zillow and a local real estate investor who buys homes) and I also got "what sales price do you recommend" solid quotes with comps from two realtors I know and trust and one random realtor. All of the "price it right" realtor pricing suggestions were completely on target with what I already knew from looking up my own comps in the area, and were about the same or even lower than the OpenDoor offer I already had! I did used my spreadsheet powers to include all costs and figure out that I would most likely lose money if I went any route besides OpenDoor.
I did some low level surface cleaning and decluttering before OpenDoor came to view it, but I didn't paint or get new carpet or stage of anything (they don't care about that.) Some areas were in great shape and some were terrible, like the wall behind the stove that was half unpainted because we took out the giant hood a year before. They had already made me an offer within a day or two of my initial online request, which was actually a fair valuation based upon subsequent data I collected, and then they only reduced their offer a tiny bit after their walk through ($2K lower, which I successfully appealed, so it was back to a full offer!) It's normal for buyers to come back and try to lower the sales price after inspection! They do charge a "real estate commission", which is how they make their money, but I ask them to lower that percentage by a point, telling them I had a new roof and new siding, and they did! I went from contacting them to having a closed sale and a six figure check for my sales proceeds in my bank account in about a month. Easy peasy!
OpenDoor did offer to rent it back to me AFTER closing (so I already had the sales proceeds), but I decided to just move out. It then took them over 3 months to sell it, which is not terrible but it would have stressed me out because it didn't sell until January. They sold it for 3% more than they paid me. Good for them! They also painted it, trimmed up my absolutely out of control bushes in the front and back, deep cleaned, mowed the whole time, etc. So, it wasn't pure profit for them, and it wasn't very much money in the grand scheme of things. And, honestly, it was worth it to me to just have the money fast to make the giant down payment on my new home, which I absolutely love!
Stop over thinking this. You need to sell your house and you need to move to a very specific location.
You are putting way too much drama into renting. Yes, the first place sounded bad, so I'm glad you bailed on that one, but most places will be okay so just find a home or rental in the location needed for the elementary school and move. Seriously. I once had to help a relative find a place to live after a home fire. We found a reasonable place to rent and he was living in the new place within 4 days of the fire. He had almost nothing to move (fire sucks), so he bought a new mattress the day he picked up the keys and spent the next month furnishing it beautifully with a selection of furniture in his preferred style off of Craigslist and art from the local thrift shop. He only lived there for a year, but it was exactly what he needed at the time. So your hemming and hawing just seems ridiculous, in comparison. Yes, this process is going to cost you some money, whether it is hiring painters or renting a storage unit or paying movers. Relocations cost money. But, remember, time is money as well! If you try to nickle and dime too much, then you will drive yourself crazy.