Dunlop K627 is an old-compound, nylon wrap on nylon and steel side chord, bias-ply tyre. Its a K591 with white lettering and been around a long time now specifically redesigned for low-speed scooters/bikes. About the same quality as Cheng Shin.
This tyre provides a soft ride due to the nylon ply. As TT says if the chords have stretched, it will permanently develop a flop.
My suggestion is
underinflate to about 25 PSI, then go for a 25 mile ride (just not too fast). The tyre should be nice and hot when you return. Elevate the wheel and wait until cooled down. Flop should be gone. However, a new tyre in probably in your future...perhaps...Bridgestone Spitfire? newer compound, bias ply, soft but grippy ride, similar value priced. or arguably, how about the best Bridgestone bias-ply, their BT45?

Edit: Don't get me wrong - I have no particular interest in Bridgestone, just comparing to a comparable design more modern tyre.