Can a cervical herniated disc in the neck cause slurred speech, difficulty talking, and swallowing? sometimes choking on thin liquids?

No: Cervical discs classically cause radiation to the upper extremities.

No: Those functions (swallowing, talking, etc) are all mediated from above the c-spine. Those symptoms are worrisome for stroke and evaluation should be sought on an emergent basis.

Evaluation needed: Swallowing problems are most commonly related to neurological disease. The most common reason for these symptoms would be a stroke in the brain stem but other reasons need to be investigated for an appropriate treatment plan to be set.