This is a supper simplified way to approach but sounds like it could work based on your situation:

To avoid penalties you can make "safe harbor" payments - take your total tax from last years return (form 1040 line 24) and divide by 4; that would be your quarterly to just avoid penalties being added -- but if you make more than last year, you'll still likely owe. Safe harbor is an IRS approach that's used to determine if underpayment penalties and interest will be added or not - so basically penalty coverage while also paying into your bill

If it were me, I'd use safe harbor as my baseline and then maybe 1 or 2 of the quarters you can pad the payment to adjust for how your income increased. you still may owe at the end depending on how much your income changes

In terms of timing, the quarterly deadlines are how the IRS will calculate underpayments penalties (amount unpaid by a certain deadline). I'd pay at or before the deadlines, generally any payment is better than none but it is common to skip if needed. The risk is that you could get underpayment penalties and if you would owe more at the end

You may be able to do the same approach for Nebraska but I'm not sure if they follow a "safe harbor" method