Wow, way to kill it! $19K in 6 months on that salary is seriously bad-ass! You are also dead on on the next steps: emergency fund and increasing Roth/403(b). My recommendations:
1. If you are not yet getting the full match, up the 403(b) until you get to that point. If someone is going to hand you free money, it's in your best interest to let them. :-)
2. Next priority is to fund a Roth for 2017, for two reasons: (i) Roths are use-it-or-lose-it (if you don't put in the $5500 this year, you can't make up for it by putting in more in later years); and (ii) you can withdraw your contributions from a Roth at any time if you really need it, so it can substitute as an emergency fund for a few months or a year.
3. Then build your EF outside of your Roth. Once you do that, you can switch your Roth over to a broad index fund (since you no longer need it to double as an EF).
4. Once you have your EF set and your Roth fully funded, start throwing the extra money into your 403(b).
Since you have $2K/mo. to play with, you can accomplish multiple goals at the same time. For ex., you have about 6 months until the April 15 2018 deadline for a 2017 Roth, so you can put $1K each month into a Roth ($500 for the final month) and $1K into the EF. Just call Vanguard and open two accounts, one for the Roth, and one for a taxable account. For now, put all the $ in both accounts into a money market fund.
Then in April, you will now have $5500 in a 2017 Roth and $6K in the EF. Now you stop the EF. And you do two things with the Roth: first, you sell the money market shares and invest them into VTSMX. Second, you also now need to readjust how much you are sending in to fully fund the 2018 Roth -- just do the math for whatever the 2018 limits are and set up automatic withdrawals from your bank account into your Roth divided by the remaining months in the year.*
At this point, you will also have more free cash left over to invest, because you'll only need like $500+/- for the Roth each month. So now you go in to HR and max out your 403(b) (or as close to maxed out as you can afford). Choose a broad index fund with the lowest expenses.
Then all you need to do every January is re-adjust your automatic withdrawals based on the annual limits and the amount you have to invest.
Two other things I'd encourage you to think about:
1. Do you have disability insurance? $5K won't go very far if you are out of work for months or years, so that is good protection while you are building up assets that could carry you through on your own.
2. After you do the Roth and EF, think about whether you need/want to save for longer-term expenses, like major car repairs or a new-to-you car when this one dies, moving costs if you need to move somewhere, travel if you want to take a big trip, etc. $5K is a decent EF, and with your expenses you don't need huge amounts sitting in cash, but it isn't enough to serve as both an EF and cover those kinds of big, periodic expense.
But those are relatively minor -- you are really doing awesome things. Congrats -- it just gets better from here!
*I prefer to do my Roths on a calendar year -- I know I have until April, but I don't want to forget or get confused, so I just go from Jan-Dec. So if it were me, I'd divide whatever the 2018 limit is by the remaining 8 calendar months to figure out the monthly contributions, but you can obviously do it either way.
