It is an electric fuel pump but seems to be working fine after I fill up with gas.
Where is it mounted and how is it wired? And what type is it?
Just wondering out loud it if's up high and relying too much on the siphon-effect to get gas to it. Most (all?) electric pumps are more efficient at pushing fuel than they are at pulling it. So those mounted towards the front of the vehicle, and/or up high, can have issues.
There are a few members here that are very lucky and their pumps mounted up on the wheel wells and firewall work fine. But in most cases, that does not work for long. Been tried, found wanting...
Is your Bronco a '77? If not, and I think you said it's a '68(?) then you'd be the odd exception to the tank size rules. The '77 was the only officially 14gal version and all the rest were somewhere around the 10-12 range. Some books show them varying by years, while others have them all pegged at about 11. My '71 was always exactly 11 when I drove it to bone dry.
But even that 3gal difference does not explain away your issue. However what the others have already said would. The pickup tube not reaching the bottom, the size of the tank being compromised, the pump mounted in a bad place or some variation of those thoughts.
And that assumes you can fill yours up fully, which many can not do.
Is it a stock tank?
With an assumed 11gal for now, filling it with 6gal means that you're leaving 5gal in the tank when you start having trouble. This might be consistent with the sock filter going bye-bye, or the metal tube rusting off part way. I've seen this multiple times where the pickup tube can no longer reach anywhere near the bottom of the tank.
If none of the other stuff pans out, maybe pulling the tank is in your future to check inside.
Paul