Update a jailbreak iphone 4

Hello. I have an iphone 4 on v4.2.1 jailbreak, bb 03.10.01 locked, and have been using several gevey cards for the past 2 years. However now, fewer and fewer apps come out on 4.2.1 and updates are also limited. My question is, do I have to remove jailbreak/cydia before I update/restore to 5.1.1 ? I have been trying to do the correct process of preserving my baseband with custom ipsw (on PC), everything is going ok, until my iphone tries to verify with apple. Then the process is stopped and (thank god) my phone is at least working as before trying this. Is there a way to go to 5.1.1. without lossing my jailbreak and my unlock status? Can I bypass the verification, should I try this without any internet connection, or will that cause other problems. I don't care if I need to get a better gevey or do something else (as long as it doesn't cost me too much). I want my iphone jailbreak and unlock status on v.5 or later. Anyone that can help please? Thanks in advance