I have fine hair too (with a mix of porosities), and I absolutely love avocado oil because my hair/skin absorbs it more readily than it does olive oil (and I like cooking with it more too!). I picked up a bottle at the grocery store, but it looks like it's also sold on the site you linked.
Another oil I've had good luck using is tamanu oil - I use this on my face/scalp when it it was super dried out, though I haven't tried this on the hair itself yet. Unfortunately, this one isn't on the site you linked
If you have a Whole Foods nearby, you may want to go and test the Aura Cacia carrier oils they have on display. They have tamanu, aragan, baobab, macadamia, and rosehip oils, which I think all fit what you were looking for. Most stores have testers out for all of these
As for the sunflower oil: it's made by pressing sunflower seeds, so I wouldn't risk it!