"Cut their hawsers"

This citation (found in the OED) shows that the term hawser was not only used for the strong ropes used in towing and anchoring, but was also used to refer to the running rigging:

?1615   G. Chapman tr. Homer Odysses (new ed.) ii. 609   With well-wreath'd halsers hoise Their white sails.

Of course, it's possible that Chapman had confused hawsers with halyards—but if Chapman could be confused in that respect then so could Daborne. Landlubbers are often confused over naval terminology! The confusion might easily arise because of two naval tactics that both involve cutting ropes:

1. To quickly disable an enemy ship, cut the rigging that supports its masts and controls its yards and sails. Unsupported, the masts cannot bear any press of sail, and it takes much time and labour to repair the cut ropes.

   For example, Captain Trollope of HMS Glatton wrote in his report on the action of 15 July 1796:

   I have no doubt, from the apparent confuſion the enemy were in, we ſhould have gained a deciſive victory, but, unfortunately, in attempting to wear, we found every part of our running rigging totally cut to pieces [i.e. by the enemy's cannon-fire], and the major part of our ſtanding rigging, every ſtay, except the mizen, either cut or badly wounded, and our maſts and yards conſiderably damaged.

2. If a ship is anchored but needs to make way urgently, then the sailors will cut (or slip) the cable (or hawser) and run. This operation saves much time and labour, but leaves the valuable anchor on the seabed.

   For example, Captain Porter of the USS Essex wrote in his report on the surrender of the ship in the battle of Valparaiso, 28 March 1814:

   I ran close into a small bay, and anchored within pistol shot of the shore [...] The enemy soon repaired his damages, and made a fresh attack with both ships on my starboard quarter, out of reach of my carronades, and, where my stern guns could not be brought to bear—he there kept up a galling fire which it was out of my power to return. The only rope not cut was the flying gib halliards, and that being the only sail I could set, I caused it to be hoisted, my cables to be cut, and run down on both ships, with an intention of laying the Phoebe on board.

OED's definition for hawser is...

A large rope or small cable, in size midway between a cable and a tow-line, between 5 and 10 inches in circumference; used in warping and mooring

I confess I didn't know what warping meant in this context, so I had to check warp...

Naut. A rope or light hawser attached at one end to some fixed object, used in hauling or in moving a ship from one place to another in a harbour, road, or river

OP doesn't make clear exactly what the current position of the "disabled" ship is, but perhaps it's such that it would need to be hauled into a different orientation in order for its sails to catch the wind and give pursuit.

Of course, not every playwright is a knowledgeable seaman. Perhaps Robert Daborne simply knew that hawsers are "ropes", and what he was trying to say was the sail ropes (sheets) were cut. Note Oldcat's comment below, and particularly note that "sheets" as used to let out and control sails are sometimes chains rather than ropes. The stretch from mooring hawser to sheet-controlling chain (or cable?) doesn't seem too great to me.

Since the etymology of hawser points to the old French « halcier » and « haucier » meaning "hoister" or "to hoist", I suspect that they are referring to what we call halyards today.

The current French word « hausser » means to raise or elevate. Hausser looks like a loan word that became hawser, where only the spelling has changed.

Raising and lowering a sail is sometimes called hauling up or hauling down. Of course the halyards are used to raise the sails. On some boats, there is a line called a downhaul to lower the sail.

So I suggest that, back in the day, hawser was most likely the halyards to the sails. During the mêlée with grappled ships, it's unlikely the sails are raised. Getting under way would be impossible until they are hoisted.

Also in sailing, we haul in or haul out a sail. By this we use the sheet line to bring the sail closer to the centerline of the boat (haul in) or let it out (haul out). (Being close hauled generally means the boat is sailing close to the wind and the sails are hauled in.

So while these sail control lines have other names, they may also have names that are closely related to hawser or to the act of hoisting or hauling.