My phone is unlock and is currently connected and has service with a cell phone company. However, if I want to see if my phone is eligible with another carrier, it will tell me that it is not because it is currently connected to a cell phone company. correct?

Yes, that is correct. In order to switch carriers, you will need to disconnect your phone from your current carrier and contact the new carrier to see if your phone is eligible for their network. The new carrier may have specific requirements or restrictions for bringing in a device from another carrier.