The reasons to sell first are valid, & many people below have explained it well.
As for what to do once that happens, there are a number of options.
Hotel is one option, as you've already mentioned. Some do offer monthly stays, such as Extended Stay America, which are pet-friendly, & offer rooms or suites with full kitchens. Long-term rentals (by the week or month) are cheaper than the one-two night stay rates.
Corporate rentals are also an option - often offering furnished homes/condos for temporary rentals. See this link for info, & note that #18 mentions the category has expended in its initial scope to include clients such as relocating families:
http://www.corporatehousingbyowner.com/faq-renters
Additionally, you can search rental sites, such as Craigslist, & potentially enter into a short-term lease. Month to Month is hard to find, but 6month is common.
The benefits to a short-term renting option such as the ones above is that you can opt to move out of your house in order to list it "empty", rather than keeping it up for showings, and once your house sells & you are ready to move into your new home, you can take your time moving in. Depending on the option you choose, you can extend your stay for select periods of time.
Additionally, if you are 'waffling' between different areas to move to (this town or that one?) you can do a rental in the area you think you like best & give it a trial run, without the long-term commitment if you don't like it.
FYI, not sure where you are looking to relocate, or currently live ("Chicago" is a big area) but I do know of a spacious condo just north of the Loop, 2 min from I94 that is coming up on the rental market soon. You can message me privately if you would like more information.
Good luck, & keep us posted. I appreciate the learning experience being shared as you go through all the decision making in the process! T. =-)