Inadequate grounding of sub panel
Summary: a ground rod will not substitute for the neutral/ground connection back to the main.

To be clear, your third wire is actually providing two functions: neutral and safety ground. The power disturbance is due to the failed neutral. 120 v loads on only one leg do not have a return path and so will not work. If you have loads on both legs, they will conduct through each other but, depending on the ratio of the loads, the voltages will vary between nearly 0v and nearly 240 volts and may destroy equipment or cause a fire.

The safety ground has multiple functions but significantly is to prevent shock with metal-cased, three-prong equipment. If a hot wire were to come lose and contact the metal case, excessive current would flow to ground and trip the breaker on the hot. If a GFCI is in the circuit, the trip will come with much less current. This prevents fatal voltages from appearing on the case.

In this case, because the neutral and safety ground are combined and have become disconnected, you are actually in worse shape. Voltage will conduct through switched on loads and actually energize grounded objects (metal-cased 3-prong equipment, outlet and switch screws, etc.) providing a great shock hazard.

This is why the electrical code has been changed to require a 4-wire feed to sub panels.

A ground rod would mitigate this to some degree but dry dirt is a poor conductor. It is a poor safety ground and a lousy neutral. You absolutly need a working neutral/ground back to the main panel.

At a minimum, I would fix the open conductor and add a ground rod at the garage. You should also consider upgrading your feed to 4 wires to separate the neutral and ground, as current code requires.