I can't find evidence of the origin of 'drag' from German dreck. Actually most sources cite its as likely origin the Proto-Germanic dragan/dragana (draw-drag) related to Low German dragge (drag-anchor). "(But not considered to be directly the source of Latin trahere)"

Bedraggled:

• limp and soiled as if dragged in the mud; "the beggar's bedraggled clothes"; "scarecrows in battered hats or draggled skirts"

Etymology references:

Etymology:

drag +‎ -le

• From Middle English draggen (“to drag”), early Middle English dragen (“to draw, carry”), confluence of Old English dragan (“to drag, draw, draw oneself, go, protract”) and Old Norse draga (“to draw, attract”); **both from Proto-Germanic draganą (“to draw, drag”), from Proto-Indo-European dʰerāgʰ- (“to draw, drag”). Verb sense influenced due to association with the noun drag (“that which is hauled or dragged”), related to Low German dragge (“a drag-anchor, grapnel”). Cognate with Danish drægge (“to dredge”), Danish drage (“to draw, attract”), Swedish dragga (“to drag, drag anchor, sweep”), Swedish draga (“to draw, go”), Icelandic draga (“to drag, pull”). More at draw.

I can't find evidence of the origin of 'drag' from German Dreck. Actually most sources cite as its likely origin the Proto-Germanic dragan/dragana (draw-drag) related to Low German dragge (drag-anchor). "(But not considered to be directly the source of Latin trahere)"

Bedraggled:

• limp and soiled as if dragged in the mud; "the beggar's bedraggled clothes"; "scarecrows in battered hats or draggled skirts"

Etymology references:

Etymology:

drag +‎ -le

• From Middle English draggen (“to drag”), early Middle English dragen (“to draw, carry”), confluence of Old English dragan (“to drag, draw, draw oneself, go, protract”) and Old Norse draga (“to draw, attract”); **both from Proto-Germanic draganą (“to draw, drag”), from Proto-Indo-European dʰerāgʰ- (“to draw, drag”). Verb sense influenced due to association with the noun drag (“that which is hauled or dragged”), related to Low German dragge (“a drag-anchor, grapnel”). Cognate with Danish drægge (“to dredge”), Danish drage (“to draw, attract”), Swedish dragga (“to drag, drag anchor, sweep”), Swedish draga (“to draw, go”), Icelandic draga (“to drag, pull”). More at draw.