The spruance class comes to mind as being armed with missiles, yet not receiving the DDG moniker, just DD.
What sets them apart is a lack of air defence as compared to their contemporaries. They are barely able to defend themselves much less any other ships, whereas pretty much all the other USN destroyers could defend other ships as well.
Their intended role at design was for anti submarine duties as carrier escort, which would have put them under the defensive umbrella of long island, virginia, and/or ticonderoga class cruisers.
Even when refit with 24 mk-41 vls cells to replace their 8 cell armored box launcher, they didn't get the DDG moniker, because they only gained more tomahawk and ASROC.
As to rocket specific armaments, there are a couple, though pretty much all are either amphibs or inland patrol craft. A notable exception are 4 project 206 torpedo boats, shtorm class, owned by egypt, which were refit with 40 round rocket launchers - though again, its intended role is shore bombardment.
Likewise, any ship with the italian Breda 105mm SCLAR is capable of launching barrage rockets having a range of 11,000m, though once again, the intended role is amphib support. Its been carried by Argentinian Meko 360H2, German Brandenburg and westerwald classes, Italian Contedi cavour, horizon, soldati, Cdte. Cigala Fulgosi, Garibaldi, audace, maestrale, artigliere, alpino, sangeorgio, stromboli, and etna classes, Nigerian Aradu class, Peruvian lupo, and Venezuelan Lupo class.
Afaik, that's pretty much it though. So no ships I am aware of in the modern era outright dedicated to the rocket attack role. Its always secondary, supporting, and especially not anti-ship. Though I suspect they'd effectively mess up smaller ships like project 206 missile/torpedo boats and various FAC.