due to being the only surface ships in the Navy aside from the old converted battleships capable of engaging a heavy surface group.
Big guns and missiles =/= ability to engage a heavy surface group. Sure, it helps, but with the exception of the Iowas all U.S. ships from about 1975 onwards were armed with, at best, medium caliber guns and rather dinky missiles (compared to their Soviet counterparts). Yet even without carrier support a surface action group of Ticos and Burkes, for example, can engage and at least hamper a heavy surface group, especially one composed of older ships - it can lob Harpoons and damage if not outright sink escorting vessels, wound larger ships and, most importantly, deny the enemy force offensive capability by shooting down their missiles.
...and I'm sure erik_t is going to come in here and prove me wrong, but I know he'll agree with this sentiment: it's kinda tiring to see people think that they need big guns and missiles to have a useful warship. There is almost nothing as a useless warships (the white elephant status notwithstanding). This really isn't meant to be directed at you, just uhhh...you were kinda the victim of a random rant here :p
Other than that...you are indeed only going to be fielding DASH on a pad that dinky. And not when SeaSlug/whatever ASuW missile is active.
Your backstory does highlight the transitional period when missile technology was maturing, so that's another extra consideration.