Do you have to keep the fans? If not, just have the electrician remove them when they are installing the recessed cans. If you need the fans, the electrician can help you place the cans so that the light will not be interrupted by the fan.

Or you can swap the fan with a ceiling light fixture.

While I'm not 100% dead set on keeping the fans, they are really helpful with air circulation. We are going to do the work ourselves with the help of my partner's dad, who is a master electrician but a far cry from an interior designer, so while he will be helpful with making sure all of the wiring is safe and to code, I don't know how great he will be at helping us lay out the position of the cans - we kind of need to have a plan before he comes over to help us.

You can get a battery operated puck light, and hold it up to position to see if you get any interference from the blades. If you keep a distance away, I'm sure it will work with the fans. Or, get a light kit for the fan.

I would try to solve your lighting needs with portable table and floor lamps first, especially in the bedrooms and living room. Just using overhead lighting from recessed cans is not good lighting design - it's not very flattering and looks more institutional.