Am I actually going to escort any ships in my escort? So far I have had 1 mission where I've done this.

I'm acting more like a destroyer than an escort.

Short answer no you will not be doing escort missions. Their called "escorts" because when the Defiant Class was introduced it was considered politically incorrect for Starfleet to have warships and be identified as such. Just a guess, but I believe "escort" is derived from "destroyer escort." Which today would be considered either a destroyer or frigate.
Regardless, escort implies a ship who's sole purpose is to fight protecting other ships- which is basically what they do in STO.

USN site:

Destroyers - DDG

Description
DDG 51 and DDG 1000 destroyers are warships that provide multi-mission offensive and defensive capabilities. Destroyers can operate independently or as part of carrier strike groups, surface action groups, amphibious ready groups, and underway replenishment groups.


Frigates - FFG:

Description
Frigates fulfill a Protection of Shipping (POS) mission as Anti-Submarine Warfare (ASW) combatants for amphibious expeditionary forces, underway replenishment groups and merchant convoys.


So our escort class is more akin to destroyers than frigates. “If it walks like a duck, quacks like a duck, looks like a duck, it must be a duck”. I guess I'll do what I usually do when Star Trek makes mistakes and mentally jigger the type to destroyer. Thanks

That's true in the US Navy, but the definitions have gotten muddled for most modern navies.
When I see "escort" I always think "Destroyer." Which fits. The other thing that the PTB may have been calling back to was the TNG Technical Manual. According to that Enterprise-E was classified as an Explorer, not Heavy Cruiser like the Enterprise-A. So according to that scheme a starship is labeled according to it's primary roll. Thus an escort is a ship designed for combat that escorts, and protects other ships. Given that the best defense is a good offense the ships are designed for speed and firepower.

Bringram pretty much nailed it. Escort in Starfleet terms was their way to skirt around the Klingons and Romulans throwing a hissy-fit that Starflet was building a dedicated warship--something the Defiant very, very much was..

A more accurate role for the Escorts in STO are that of the Panzerschiffes, or the German "pocket battleships". They've got the firepower to take out almost anything and the speed/mobility to run away from anything they can't engage directly--which isn't much...