I used to prepare my own taxes before I had investments. I've also been a single mom looking for any and all tax breaks.
You don't get to claim dependents on your kids just because they are dependant on you out of your allowing them to be. :-) They would have to have some mental or physical disability to an extent the IRS says so. .
If your son was full time student at any point in that year, you should check to see if that helps your case.
Just a thought: If you are low income and you have able bodied children that won't go to school full time, it may be time to give them some education on what kind of salary they need to get by in life for real. Perhaps they will change their minds about the need for school or technical training. You are the captin at the wheel here. You might have to try some "tough love" for all your benefits and have them pay rent and part of utilities and groceries if that helps you. If not, if it helps you, they might need to move out.
As to not having to pay your taxes if you delay your Return, not so. They want you to pay what you legitimately think you owe by the usual deadline or there can be fines and fees that mount up quickly. You can ask them about a payoff plan in case there are new rules.
But for now, I'd consider you and your kids pinching your pennies til then and together, coming up with what you owe. Its time for payback from your giving them free room and board.
To avoid this from happening next year, review how many deductions you take on your payroll. I'll bet you didn't change them when your kids became 18 and not full time students. If you own your home, that gives you tax deductions. If you rent, nada. Rents are higher nowdays. Consider being a homeowner if you are not (even if you have to downsize to a condo or townhouse to avoid having to pay all repairs, yardwork, etc. Consider having a side business that makes money so you can take deductions from your related expenses which can also help your domestic side (like paying for the lease payments on the car you must have for a business purpose).
If you are using Turbotax which prompts you to deductions, I presume you took all that you could. If unsure, try a tax service one year and ask abundant questions about whether you missed anything or could create more deductions next year from actions you take now.