It is difficult to say what the root cause of your problem is. This is because I can't figure out why your hands would feel swollen without begin able to see a difference in the way they look. Perhaps the swelling is so little that you can't see it.
Strictly speaking, you dad is correct. Eating a high sodium diet causes water retention which can in turn cause swelling. However, people that have symptoms after eating a high sodium meal often have something wrong with their heart or their kidneys. This is because the body is designed to get rid of the extra sodium in the urine.
To address your last question, it is always a good idea to reduce the amount of sodium in your diet. This is because reducing the amount of sodium you eat can reduce your blood pressure. I do think that you should see a doctor
about your hands and feet swelling. You could start by scheduling an appointment to see your primary care physician
. He or she will listen to your symptoms and perform a thorough physical exam. If no swelling is found, then your doctor may just give you reassurance that everything is fine. If swelling is found, then your doctor may do some blood work to look at your kidney function.