Askamanager.org would say that a gap of less than 6mo is no big deal to most employers and won't require "an explanation" as long as you have an otherwise solid work history. And, if an employer DOES demand an explanation, it's probably not a company you want to work for anyway.
Start looking in June-ish, and file for unemployment then, when you can be honest about your search. You probably wouldn't have an offer before August anyway. If you want, tell the employer that you're currently taking care of a family member, but you expect the situation to be resolved by [Aug/Sept, whenever school starts] so a/n [Aug/Sept/Oct] start date would be ideal. Remember, withdrawing yourself from consideration for a job is NOT the same as turning down an offer.
Just put the end date with your recent employer on LinkedIn sans explanation. You can explain the circumstances in the interview. I don't think employers will automatically assume you were fired, especially if you have a reference from the company.
If there are appropriate employers in your close geographic location, I would target them first. Think about how much more money would be necessary to make a commute worth your while when you apply to farther-flung employers and build that into your required salary range for the positions.