Negril is certainly different. With all the "entertainment" (bars, clubs, etc.) closed by order and most small businesses closed by choice, walking the West End is a new experience.
The beach was only closed on the Easter Weekend and with nearly every hotel not being open makes for another unique walking experience.
We know several teachers, but virtually no one is allowing strangers in their house at this time which is the correct thing to do.
We helped as many visitors as we could to find new flights when theirs were canceled and they were happy to be returning home. For those with scheduled flights after the flights were grounded, many had to seek new lodging when their hotels shut down. We directed them to the remaining open hotels.
With that in mind, the only option I can think of for you to find Internet access would be to try one of the two hotels that we know have been open. The closest for you would be Seastar Inn. I am not sure how or if they are charging for access, but buying something there may get you access for the kids.
As for your reemployment websites, there is a strong chance that being in a foreign country is blocking the access. You will need to download a VPN that makes your device seem to be in your home country.
Of the few dozen visitors who are still "stranded" here, they worked out their Internet issues early on and have stayed in touch with their family and friends. I remember you asking about getting a new SIM card sometime last year, if you have a smartphone, buy credit and get a plan with lots of data. Turn on your hotspot on the phone and you should be ready to go. Your kids can connect to your phone just as they would any other WiFi.
If you have any other questions, feel free to ask.