TLDR: When shopping rates you need to be looking at what the title fees and origination charges are, not the total amount of "stuff" you have to pay at closing, some of those a costs are already baked in to your current loan and it's a like for like out of pocket. You also only want to compare principal, interest and mortgage insurance payments, and back out tax and insurance payments as they're going to be the same.
Longer explanation: Escrows are not closing costs and I would not roll those into the loan. You have the cash in an account on your current loan, why would you want to pay interest on that for 30 years? You're also rolling in a month of escrow payments when you're going to be skipping at least one, if not two payments. Float the cash flow for 30 days if you can, your lender has to refund the balance in that time frame.
Your $1600 payment is not what you use as a basis for comparison, that includes taxes and insurance and any escrow shortage make up. You want to look at the cost of the debt, which is the principal, interest and MI payments (if you have any) on the current loan against the new loan to determine your actual payment reduction.
I'm also guessing that your LTV is going from either 80% on the current loan (No MI) to higher than 80% (MI) or from above 80% LTV to and even higher LTV which puts you in a higher risk category, which means higher MI payments. This is based upon the recency of your purchase, because most people do not put more than 20% down on their house.
Your loan officer isn't doing you any favors by not explaining any of this and just wants to get you out the door happy that you're "saving" $30/mo. Absent all that you're also probably paying points to get to that 2.875 based upon the netted out cost of the loan (12,500 minus 6,000 escrow, minus another 800 for a month of escrow, minus a month of interest for 500 is $5200 which seems outrageous on your loan balance). Maximizing your savings is part of their job if they're worth a damn. I'd tell that person to piss off and find someone that will actually put in the effort to help you, they're making enough off you to do that.
If you can't go out of pocket for your closing costs you're probably not going to be able to see the benefit of refinancing, because between loan balance increases, LTV increases and subsequent MI increases you're going to eat up all your monthly savings. You should see a ~$140/mo drop in your payment given your loan balance and rate reductions, which is worthwhile if you can find a place with fewer fees and can cash flow the up front cost for a month.
3200 in closing costs (which is still probably high) and $140/mo reduction is right where you want to be as far as refinancing.
