The whole Italian surrender thing was badly mismanaged, probably due to a fear of giving the Germans advance confirmation of the Cassibile armistice.
The first SS panzer korps was stationed in northern Italy at the time (recovering from Kursk) and did take part in rounding up and disarming Italian troops, how the SS would have reacted to masses of escaped POWs is anybody's guess but it could have turned really bad.
The partisans were not strong enough in 1943 to be much help, there were no partisan "sanctuaries" in 1943 the POWs could escape to.
On the other hand the thought of thousands of escaped POWs moving around in their rear may have well convinced the Germans to retreat from southern Italy as they had originally planned, Rommel's analisys considered a stand south of Rome as unfeasible, only after Kesselring's stand at Salerno proved him wrong was the strategy changed.