Does falling asleep constantly during the day and feeling tired mean that I have narcolepsy?

Not really. Typically, it means you are not having restorative sleep at night.
Maybe. The most common cause for sleepiness duringthe day is poor night sleep and, too little sleep. Sleep apnea, a breathing problem at night can cause severe sleepiness. And, yes narcolepsy. You need an overnight sleep study to test for sleep fragmentation. You need a daytime napping test (mslt), if the overnight sleep study is normal. A sleep neurologist can sort this out.