91 LX issues

If it cranks over slow and you know the battery is strong, I would check/clean up the ground wire where it oonnects to the block. When I got my 90, the ground lead was attached to the timing chain cover. Once I moved it to factory location on the block, no more issues.

Never, never disconnect an alternator from the battery with the engine running. The resulting voltage spike can damage the car's electronics including the alternator.

Alternator troubleshooting for 86-95 5.0 Mustangs:
Do all of these tests in sequence. Do not skip around. The results of each test depend on the results of the previous tests for correct interpretation.
Changes in wiring for a 3G alternator are in red.

Engine off, ignition off, battery fully charged.
1.) Look for 12 volts at the alternator output. No 12 volts and the dark green fuse link between the orange/black wires and the battery side of the starter solenoid has open circuited.
3G alternator: Look for 12 volts at the stud on the back of the alternator where the 4 gauge power feed wire is bolted.
No voltage and the fuse for the 4 gauge power feed wire is open or there are some loose connections.
2.) Look for 12 volts on the yellow/white wire that is the power feed to the regulator. No 12 volts, and the fuse link for the yellow/white wire has open circuited.

Engine not running, ignition on, battery fully charged.
1.) Alternator warning light should glow. No glow, bulb has burned out or there is a break in the wiring between the regulator plug and the dash. The warning light supplies an exciter voltage that tells the regulator to turn on. There is a 500 ohm resistor in parallel with the warning light so that if the bulb burns out, the regulator still gets the exciter voltage.
Disconnect the D connector with the 3 wires (yellow/white, white/black and green/red) from the voltage regulator. Measure the voltage on the lt green/red wire. It should be 12 volts. No 12 volts and the wire is broken, or the 500 ohm resistor and dash indicator lamp are bad. If the 12 volts is missing, replace the warning lamp. If after replacing the warning lamp, the test fails again, the wiring between the warning lamp and the alternator is faulty. The warning lamp circuit is part of the instrument panel and contains some connectors that may cause problems.

2.) Reconnect the D plug to the alternator
Probe the green/red wire from the rear of the connector and use the battery negative post as a ground. You should see 2.4-2.6 volts. No voltage and the previous tests passed, you have a failed regulator. This is an actual measurement taken from a car with a working electrical system.

Engine running, Ignition on, battery fully charged:
Probe the green/red wire from the rear of the connector and use the battery negative post as a ground. You should see battery voltage minus .25 to 1.0 volt. If the battery voltage measured across the battery is 15.25 volts, you should see 14.50 volts

3.) Familiarize yourself with the following application note from Fluke: See Automotive Test Tools for help for help troubleshooting voltage drops across connections and components. . You will need to do some voltage drop testing of several of the wires.
Start looking for these things:
A.) Bad diode(s) in the alternator - one or more diodes have open circuited and are causing the voltage to drop off as load increases. Remove the alternator and bench test it to confirm or deny this as being the problem.
B.) The secondary power ground is between the back of the intake manifold and the driver's side firewall. It is often missing or loose. It supplies ground for the alternator, A/C compressor clutch and other electrical accessories such as the gauges. Do the voltage drop test as shown in the Fluke tech note link. Measure the voltage drop between the alternator frame and the battery negative post. Watch for an increase in drop as the load increases. Use the Fluke voltage drop figures as guidelines for your decisions.
C.) Bad regulator that does not increase field current as load increases. Remove the alternator and bench test it to confirm or deny this as being the problem.
D.) Bad sense wire - open circuit in sense wiring or high resistance. The yellow/white wire is the voltage sense and power for the field. There is a fuse link embedded in the wiring where it connects to the black/orange wiring that can open up and cause problems. Disconnect the battery negative cable from the battery: this will keep you from making sparks when you do the next step. Then disconnect the yellow/white wire at the alternator and the green fuse link at the starter solenoid/starter relay. Measure the resistance between the alternator end of the yellow/white wire and the green fuse link: you should see less than 1 ohm. Reconnect all the wires when you have completed this step.
E.) Bad power feed wiring from the alternator. Use caution in the next step, since you will need to do it with everything powered up and the engine running. You are going to do the Fluke voltage drop tests on the power feed wiring, fuse links and associated parts. Connect one DMM lead to the battery side of the starter solenoid/starter relay. Carefully probe the backside of the black/orange wire connector where it plugs into the alternator. With the engine off, you should see very little voltage. Start the engine and increase the load on the electrical system. Watch for an increase in drop as the load increases. Use the Fluke voltage drop figures as guidelines for your decisions.