Facetime, iMessage, and Mail not working after iOS 6.1.3 update

Question

I've had my iPod touch 4th generation since Oct 2010 and never updated the iOS until very recently. Before the update, Facetime worked consistently and easily. After I finally got around to updating my iOS a week ago, my iPod has come into severe issues.
99% of the time, I cannot send or receive Facetime calls. If I try to make a call, it doesn't make the dial tone and eventually it just says "Connection Lost". Nothing happens if someone is trying to send me a call. I only Facetime with my boyfriend and his iPod hasn't changed since Facetime was last working. HIs other iPod is also updated with iOS 6.1.3.
I never used iMessage before the iOS update, but because it says on Apple support that it uses push notifications like Facetime does, so I tried it. Most of the time, I cannot send or receive messages. I rarely used Mail before the update, but I never had a problem with it. Now, when I enter my gmail password, it keeps saying my username and password are not correct, even though I haven't changed my credentials.
I only use my iPod for Facetime, iTunes, and Pandora. I don't need iMessage or Mail, but I am testing with those to see if this can help find the root cause of the problem. Pandora works smoothly so I know it's not a wifi connection issue. I have searched for remedies on end and tried:
1.  Reset, turn off/restart, turn Facetime on and off, etc.
2. Changing the DNS server address to 8.8.8.8, which made Facetime work normally for one time and never again
3. Updating my Apple ID password online, then signing out and signing back in
4. Creating another Apple ID with my other gmail account. Maybe 1 out of 10 Facetime call attempts, I can generate the dial tone, but when my boyfriend clicks to receive the call, it says that I am unavailable. iMessage has about the same success rate. It seems when I miraculously can send or receive texts through iMessage, I can manage to get the dial tone when I try Facetime, though the call cannot go through.
This is of course extremely frustrating and I highly regret the update now. I was hoping for an improved iPod after the update, and now I have a near-dud! I did make sure to check my location and date and time, and both are correct (US, Central time). Any help would be greatly appreciated. Thank you.

Answer 1

I spoke too soon. I tried Facetime again last night and had the same problems. Another factory reset did not work. I think it may be time to just call Apple support.