Hydrochlorothiazide, often abbreviated as HCTZ, is not associated with causing a cough. The prescribing information for the drug does not list it as a possible adverse effect and coughing is not documented as a side effect occurring in clinical trials for the drug.
However, there is always a chance that coughing due to hydrochlorothiazide is secondary to other side effects it may cause. For example, hydrochlorothiazide is associated, although rarely, with sialadenitis which is an infection of the salivary glands. Something like this could certainly cough you to cough.
Hydrochlorothiazide is also associated with various GI effects such as cramping and irritation. If you have acid reflux, HCTZ could theoretically exacerbate the condition, which could cause you to cough as well.
Lastly, as you mentioned you have a sulfa allergy, hydrochlorothiazide should generally not be given to those with have a confirmed sulfonamide allergy. The swelling in your face and coughing may be due to a reaction you are having to HCTZ.
Taking HCTZ With A Sulfa Allergy
Per the manufacturer of hydrochlorothiazide, its use is contraindicated in patients with sulfonamide hypersensitivity, although documented and confirmed cases of cross-reactivity between sulfa allergies with antibiotics and hydrochlorothiazide are rare.
HCTZ is a "thiazide" diuretic, which are derivatives of sulfonamides but they lack certain structural components of sulfonamide antibiotics (e.g. N4-aromatic amine). Thiazide diuretics are presumed to have a lower risk of allergic reactions in patients with sulfonamide allergy due to the differences in chemical structure between them and sulfonamide antibiotics.
Nevertheless, patients with a documented sulfonamide allergy or previous anaphylactic reaction to sulfa antibiotics are considered to be predisposed for development of allergic drug reactions and therefore, use of HCTZ should be done with caution. Be sure to let your doctor know about the reactions you are experiencing with the drug, as it may be causing your coughing and swelling reactions.