I think we have fixed the fuel issue!  I rebuilt the fuel pump and blew out the fuel line from the tank to the pump. I did get a little bit of rusty crap out of the line but not what I would consider excessive. There is a tag on the fuel pump that says "rebuilt" but I don't think they did a complete job. The diaphragms appear fairly new but the valves appear to be very old. they do look anything like the new ones included in the rebuild kit. Once it was reinstalled, I dumped a little gas down the carb and it started right up ( I knew the carb bowls were full) the fuel pump sucked up the gas and filled the sediment bowl and the fuel filter immediately. the fuel filter is now all the way full, where before it would only get about 2/3 full and was very bubbly. It seems to run very nice at various RPM's. I was not able to drive it tonight due to rain.  I am waiting for morning to see if the filter is still full or if it drains back. It has been a couple hours now and it is still full so I hope that is a good sign. 
I want to say thanks to everyone that offered tips and ideas.

My theory is the valves in the pump are leaking a little and fuel is syphoning back to the tank. You start up and run on the fuel that is in the carb until it runs out. Just a thought!

Yes, and that scenario wouldn't be unusual. Modern ethanol laced gasoline is said to be the culprit, attacking fuel system components made of Buna rubber.

I just checked the fuel filter now before I started the car. The filter is between the pump and the carb, and it is empty. I know it was the normal 3/4 full last night when I shut the car off. When I start the car, the filter starts filling up. I am guessing that means there is a leak in the pump and it is unable to hold pressure. I did get a fuel pump rebuild kit and a new fuel filter, I have not installed either yet.  I just want to be more certain of the cause of this problem. This problem just started last week. A little more background-- I just got this car 2 months ago. It had been sitting more than being used, in fact, it has only had 16 miles put on it in the last five years according to receipts with miles in/out listed.  I spent the first three weeks going through things (brakes and carb) before I could really drive it. I have now put 180 miles on it. Could the fuel pump go bad now that it is being used after being used hardly ever? 
 

just a silly thought  ,,,replace the fuel filter

Is there fuel in the bowl between pump and carb before you start the car?

I had a similar problem my engine would shut off when I would go up an incline crossing a local bridge. I would go fast enough to coast over the high spot and start downhill. I have a standard shift and the engine would start right up after going over the crest. I have Holley T pot and the carb float level was set 1/4 inch to low, No more problem when float was raised up to where it should have been.

I thought of low fuel level, so I drained the tank. To my surprise I drained out 12 gallons! Like your car, there is not evidence of fuel leak, but I will check/replace the fuel line. It is just strange to me that the sediment bowl on the bottom of the pump is full, but the filter on the top of the pump is empty. I know that intermittent problems are the hard ones, so I am trying to keep my patience. Last night it did the same thing, run out of gas at slow speed in 200 yards, but I drove 5 miles at 50 mph and never had a hiccup.

Does the fuel starvation event only occur when the fuel tank is at a given level or lower?  I mention this as I had a ’55 Ford that had a crack at the top of the fuel line where it connected to the inside and front of the fuel tank that only gave a problem when the fuel level was half full or less.  It took a while to figure this one out but once it was realized that the problem did not occur with a full tank of fuel, finding the root of the problem became easier.

A '54 Ford car at a different time had a small break in the flexible fuel inlet line going to the fuel pump that gave sporadic fuel starvation issues.  That particular problem showed up so inconsistently it was a hard one to find as it was not leaking fuel at that point that was evident.  There was just enough air being pulled in at that break that it was minimizing the fuel volume.  That flexible line which resembled a brake hose was simply replaced with a section of rubber fuel hose and problem solved.

Lorena, Texas (South of Waco)

There are actually a few tests a fuel pump has to pass to meet spec - pressure is one that most are familiar with, and output volume, and then vacuum on the suction side. It's pretty easy and quick to check, a mechanics vacuum gauge makes quick work of it. Does seem like a strange problem! But I bet it will fail on the suction side?? Let us know what you find ...