Fuel gauge says full all the time ,95 cougar.

Cathy, 1996 Mercury cougar gas gauge not working. Simplest thing to check first is your fuel gage fuse, make sure all fuses are good. The ground wire could be corroded/damaged, sending unit in tank faulty, fuel level gauge in cluster faulty. All things can be tested by a good Tech.,diagnosed and estimate for repair.
1994 Mercury Cougar gas gauge don work
Ford Gas Gauge Does Not Work Help Blue Springs Ford Parts Blog
Diagnose Fuel Gauge

The fuel gauge is either a stuck needle or a stuck sending unit.

Why was the battery missing to begin with? I would start by checking all fuses under dash and Central hub. Check for burned in-line fuses,if any, coming off of battery cables.

The injector needs one side power and the other is control,or a ground the main sensor resposible for injector on time is the cam position sensor.

Most of the time its is the sending unit in the fuel tank. The test for it is to use a variable resistor that is hooked up to the wiring at the fuel tank and watch the fuel gauge if the gauge works ok thats makes 100% sure its the sending unit. At 245 ohms the gauge should read full, 180 ohms for 3/4 full, 160 ohms for 1/2 tank, 100 ohms for 1/4 tank and 42 ohms for empty. The color of wires at the fuel tank is purple wire and black wire. You can also use fixed value resisters. If the fuel gauge the wires from the sending unit to computer then to gauge for bad connections, shorts, broken wire and bad grounds. If the wires test ok that only leaves insterrment cluster or computer which even most dealers can't test eather one but the insterment cluster would be a more common problem then the computer. But about 95% of the time it will be the fuel level sensor in the gas tank that causes the problem. Good luck.

sounds like it is not hooked up or not correctly. I would double check the connection on the fuel pump. Hope this helps tim

shifts first to second hard

This is a normal concern.
The instrument cluster uses 4 different operating modes to calculate the fuel level:
Anti-slosh (default mode)
Key OFF fueling
Key ON fueling
Recovery
After a fuel fill up, the time for the fuel gauge to move from empty (E) to full (F) ranges from 2 seconds to 55 minutes depending on which operating mode the fuel gauge is in.
The default fuel gauge mode is called the anti-slosh mode. To prevent fuel gauge changes from fuel slosh (gauge instability due to changes in fuel sender readings caused by fuel moving around in the tank), the fuel gauge takes approximately 55 minutes to go from empty (E) to full (F).
The key OFF fueling mode (2 seconds to read empty [E] to full [F]) requires 3 conditions be met:
The key must be in the OFF position throughout the entire refueling of the vehicle.
At least 15% of the vehicle's fuel capacity must be added to the fuel tank.
The instrument cluster must receive a valid key ON fuel sender reading within 1 second of the key being put into the RUN position. The key ON sample readings are considered valid if the fuel sender reading is between 15 ohms ± 2 ohms and 160 ohms ± 4 ohms.
If these conditions are not met, the fuel gauge stays in the anti-slosh mode, which results in a slow to read full (F) event.
The key ON fueling mode (approximately 90 seconds to read empty [E] to full [F]) requires 3 conditions be met:
The transmission is in PARK (P).
The key is in the RUN position.
At least 15% of the vehicle's fuel capacity must be added to the fuel tank.
In key ON fueling mode, a 30-second timer activates after the transmission is put into the PARK (P) position (automatic transmissions). When the 30-second time has elapsed and at least 15% of the vehicle's fuel capacity has been added, the fuel gauge response time is 90 seconds to read from empty (E) to full (F). When the transmission is shifted out of PARK (P), the fuel gauge strategy reverts to the anti-slosh mode. The key on refueling mode prevents slow to read full events from happening if the customer refuels the vehicle with the key in the RUN position.
Recovery mode is incorporated into the instrument cluster strategy to recover from a missing fuel level input during a refueling event. Missing fuel level inputs result from intermittent opens in the fuel sender or its circuits. Recovery mode (empty [E] to full [F] approximately 20 minutes) is initiated when the following 2 conditions are met:
The instrument cluster is in the anti-slosh (default) mode.
The actual fuel level in the tank is greater than what is being displayed by the fuel gauge

This sounds like a bad ground but first check the fuse for the instruments located in the fuse box. If that is OK, somone will have to find the ground wire for the instruments and see if it has any voltage with the key on. If it does, the gound connection is bad

God Luck

your float is stuck on the fuel pump