Should I use the money to pay off student loans and future grad expenses for me?
Yes. The main drawback to student loans is that they cannot be gotten rid of except by paying them off (other than extreme circumstances such as death or complete disability). A mortgage, car loan, or other collateralized loans can be dealt with by selling the underlying collateral. Credit card loans can be discharged in bankruptcy.
Stop borrowing for college, pay for it in cash, then decide what to do with the rest. Make sure you have a comfortable amount sacedsaved for emergencies in a completely liquid account (not a retirement account or CDs), and continue to pay off with the rest. You might also consider putting some away for your kids' college, so
I want to get my older son into a private middle school for 2 years. They have a hardy endowment and may offer us a decent need based scholarship if we look worthy on paper
I have a hard time getting behind this plan with a 238K mortgage. If you want to apply for scholarships that's great - but don't finagle your finances to look like you're poor when you have a quarter-million-dollar house. If you want to save some for private school then do that out of what you have. Otherwise either rearrange your priorities so you can afford it or private school might not be in the cards for you.
That said- while it was a blessing to be able to pay off the second mortgage and credit cards, your hesitancy to pay off the student loans makes me wonder if you will start living within your means after the loans are paid off. My concern is that your current spending levels that got you in this much debt in the first place will put you back in debt in the near future, and you won't have another inheritance to help pull you out.
I know that wasn't your question, but I felt like I needed to add that to my answer as well.