You should have two savings accounts:
1) Emergency savings - This is for unemployment, medical emergency, ransom money (I kid), etc. Basically, you should not touch this money unless an actual emergency comes up. Most sources will suggest 3-6 months of living expenses in cash, which you can adjust up or down based on your risk tolerance, other liquid assets which you could use if necessary, job stability, etc.
2) Specific savings - This is for an upcoming car purchase, home down payment, appliance/tv/whatever purchase, home improvements, etc. This account should only be used for these purposes, and not be commingled with #1.
$10k vs $16k isn't really a big difference. If an even bigger cushion helps you sleep at night, then go for it! At the end of the day, we all have different goals and needs. So maybe a good middle ground is $15k in emergency savings which you do not touch unless an actual emergency comes up. Then, open up a second account which you can slowly contribute to for a house purchase/car purchase/other stuff combined. If you end up putting a down payment on a home, then wait until you can rebuild to purchase the car or buy the new stove. In no case should you touch the emergency savings account. This forces you to be disciplined.
For the 2nd savings account, come up with a list of things that may come up in the next 5 years, come up with a target $ amount, divide by # of months, and start putting money away. Any remainder of your income after expenses should be invested.
There are many ways to get by on a smaller savings account balance, but given your modest income, it's probably best to keep a cushion and not cut it too close.
Hope this helps? Good luck with the home/baby/everything else!