This is why I say to answer with just the specific symptoms and not the descriptions.
Could it be and ecu problem
Have you pulled codes? What are they? No codes? No EEC problem.
An alternator will charge with a bad diode in the regulator. It will work as advertized in every way an alternator should except for one.
With the motor off and an alternator with a bad diode, the battery tries to run the alternator like a motor. This causes battery drain. What this generally DOES NOT cause is a slow,warm starter. Generally those issues are:
A tired starter
Bad ground wires
Weak battery
What causes a weak battery? Age, environment, damage, and/or inadequate charging.
So.... Let's start eliminating possibilities.
Your battery you say is new.
Your alternator is new and has RECENTLY been checked on machine SPECIFICALLY designed to check alternators and by a qualified technician. In other words: Don't tell me you went to Pep Boys and watched the fat kid with the chili dog stains on his shirt, hook some wires to your car and shake his head while pressing buttons. I don't consider him technically qualified because he's Pep Boys employee.
In the meantime, you can begin removing those pesky 20+ year old ground wires to clean them up, inspect them, and replace as necessary.