Can I extend my Non-O visa while my husband changes his work permit and visa?

Can you clarify what you mean by "the entire process"? Where in the process will your husband stop working for his current employer and where will he start working for his new employer?

There's no such thing as a 1-year non-O visa. You might have had a visa at one point, but you're on an extension of stay now.

Is your current extension of stay based on being a dependant of your husband who has a work extension?

There's absolutely no reason changing companies would take 4 or 5 months.

If your husband plans to stop working then both his and your extension will be invalid the moment he ceases employment and he will have to cancel the extension as of that date.

If he's just transitioning from one job to another, then his work extension can remain valid as long as he changes his work permit from one job to the other seamlessly.

You extend as usual then, when the changes occur, get your visa status updated with the new work permit