Starting problem on 1987 Cadillac Sedan de Villle, V6

The motor was replaced a year or so ago, with a Jasper, and only has a little over 30,000 miles on it.
The car ran out of gas or very, very low and prior to this time, the car started and ran great. I have checked the manual that came with the car and it offers no solution.
I have checked the fuel pump and it clicks on when the key is turned to on.
The car is not flooded, but the battery tries to turn over, but it is losing charge quickly now because I have been trying to start it so much. I have tried jump starting it, but the motor won't turn over and there is a puff sound that can be heard under the hood. It is not backfiring. The noise is quieter than that.
This car is not tagged and due to the age, I do not want to spend a lot of money on it, but I would like to fis it.
Is there any certain procedure to adhere to when this model Cadillac runs out of gas?
I know the following: It is not the battery, fuel pump, starter and it has gas in it now, so can you help me?