My doctors say that taking antihistamines can make a sinus infection worse or impede it from resolving; why would this happen?

Never heard of it.  The usual antihistaminic side effect is drowsiness, a second generation antihistaminic terfenadine prolonges the QT interval of the ECG risking often fatal arrhythmias, so has been withdrawn from the market in most countries.  You should only use antihistaminics if you are allergic, or for the side effect so to be able to fall asleep. If your nose is congested but you are not allergic, using an antihistaminic is useless. Drying out the mucous membranes is not a side effect of antihistiaminic, but is well known in many other drugs e.g. tricyclic antidepressants. Inhaling hot water vapor to which dried camomile blossom has been added helps:  you put some dried camomile blossom in a basin, pour boiling hot water on it, and sit above the basin with a towel around your head to make a sort of tent to capture the hot vapor you inhale.  Camomile blossom contains so called etheric oils (oils that will evaporate) that has an anti-inflammatory effect, so is good for inflammations and infections which causes inflammation of the upper respiratory tract.

Mr. Sanchez is basically correct. The inflammatory response serves a purpose, and should be allowed. There are times, however, when the inflammatory response may become life threatening, such as anaphylaxis that can cause difficulty breathing, and even the total closure of an airway.