The two branches of this nerve are pulled out of the way during the operation. The stretching and compression are likely the cause of this usually temporary problem. But it can take months for the nerve to recover. Returning to normal may take even longer.
There are some tests that can be done to assess the extent of the damage. Two tests to measure changes in the vocal cords are phoniatric evaluation and laryngeal electromyography (EMG). For problems with swallowing, you can also have an endoscopic evaluation and videofluorography.
Each of these tests helps identify the severity of nerve damage and function. Your neurosurgeon will be able to advise you about further testing. You may just need a little more time for final healing to occur. Nerves regenerate at a very slow pace and require patience. Permanent nerve damage is possible though. The testing will help determine if that is the problem.