Sorry about your situation. The strategy with this incoming cash is to consider whether or not it will be there when you will likely need it, i.e., volatility. Of course, putting it all in (highly volatile) VTSAX and waiting 30 years will, statistically speaking, produce the best result ignoring other factors. That being said, you want to be able to weather a storm without decimating your savings in the event of, say, a job loss and a market downturn, which are highly correlated.
Do I invest all of it in VTSMX at once?
I would say "no". Fill up an emergency fund first, to hold in cash (by which I mean an instant access, high yield account, like a money market (currently returning 2%), then fill up your portfolio according to your desired asset allocation (bonds, stocks, etc.)
Do I invest $10k each month, or is that playing at market timing?
Yes, it is market timing. If you can stomach the volatility and have enough cash to ride out a downturn, you should put it all in at once.
All that being said, in your situation, given the changes going on in your life, I might be inclined to sit on cash (in a money market account) for a while until I have a clearer idea of what my post-divorce life is going to look like.