Apple is no longer signing iOS 7.0.6, so new installations of that firmware are not allowed anymore. That's why you're experiencing the 319x error.
- Stay on 6.1.x. This is what I would advise - iOS 7.0.x is terribly laggy on the iPhone 4, and iOS 6 is by far the best firmware for iPhone 4 handsets.
- Upgrade to 7.1. iOS 7.1 drastically improves performance, especially on the iPhone 4 and 4s. It is better than 7.0.x, in all ways. You can indeed jailbreak the iPhone 4 on 7.1, using a tool named Pangu. You can also use GeekSn0w for a tethered jailbreak.
- Wait for a 'signing fluke'. Sometimes, Apple randomly starts signing older firmwares for no apparent reasons. I call them 'signing flukes'. For instance, in July 2013, Apple randomly started signing 6.0.x when 6.1.3 was the latest available firmware, allowing users who had updated to 6.1.3 to jailbreak their handsets. If you are especially patient, you could wait for something like this to happen again, but don't hold your breath - they are very rare! You can keep tabs on Apple's signing statuses from icj.me.
One more thing...
I downloaded the IPSW for it and shift-clicked on update
Many people in the jailbreak community advise that you do not use update functionality of any kind on jailbroken devices, as anything could happen. Mainly, if you do this, the jailbreak data will clog up your phone, and there will be a large amount of wasted space on your phone, forcing you to restore. So, to save time, restore your phone instead of updating it!
If you want to preserve your data, you can back your phone up to iCloud/iTunes and then restore from the backup after restoring your phone. This also leaves traces from your jailbreak, but 'good' traces (like, for instance, if you used TetherMe to enable tethering, it stays enabled when you restore from a backup!)