Can sinusitis really ause facial swelling into cheeks around jaw and neck?

Absolutely. Serious sinusitis, called acute bacterial sinusitis, can definitely cause local facial swelling around the eyes. This definitely requires abx, unlike many attacks of common 'sinusitis.' But here are other infections that can swell these areas: tooth infections, salivary glands, soft tissue from bites/skin conditions, swollen lymph nodes. Majority facial swelling (50%) is dental source.