It has tenants, I have had one month of vacancy since 2012, I would prefer to sell it without an agent, however I worry that there will be dozens of investors wanting to walk through the place, disturbing my tenants, making them antsy and the investors then low ball or otherwise do not generate reasonable offers.What is the best strategy for making sure the potential buyers are serious before letting them in? 

One way or another, and if nothing else, someone has to evaluate the property as to the probable cost to bring it to an after-repair-value that matches the surrounding market's fair market value for comparable properties - before ever offering to buy it at "such a ridiculously low price" (as the seller will term your offer at first), let alone, before offering to sell it at "wholesale" to your buyer (i.e., so far below fair market price at retail that the repairs can be covered as well as your profit and the flipper's profit in its eventual retail sale -- or 'buy-and-hold-to-rent-out'er's profit before they would expect to get it rented -- plus the title search, closing, and any tax costs related to your purchase from the seller (usually split somehow) - and finally, the title search, closing, and any tax cost split with whoever buys it from you.Now, that someone could be you (you do trust you and your repair-estimate abilities, don't you?)

I'm looking at rents of $4750, which are low.  

My unit does not have its own meter and has a breaker box with most of the outlets and lights within my unit, but no access to the main breaker.

Be sure to have one installed and have it wired to the light switch, ideally with a delay so it keeps running for a few minutes after your tennat took a nice hot shower.Also, make sure you educate your tenant about mold and why it's growing.

Just to play Devil's Advocate, are you confident in vacancy being that low?

Should any warning lights be going off or it "really" doesn't matter?

As long as you surround yourself with the right people and energy, you will grow at the speed of light.

I am almost certain my offer will be declined but since the house has been on the market for over 90 days in a region that most houses sell in less than a month I took a shot at a low ball offer.

As you can imagine the low fee listing companies want easy buyers and to do volume to make any kind of return.