Wont start unless jumped, and loss of power hitting throttle when cold?

How old is your battery? And are the terminals clean and tight, no corrosion, looks like mold either black or greenish? And 'wrong' alternator, as in less amperage rating...if it had, say, a 100 amp alternator and now has an 80, make no difference. The battery can accept only so much charge rate. And is the CEL or charging system warning light on? Many cars have an icon the looks like a battery, meaning the battery is not taking re-charge after start

No all terminals are clean no warning icon and I'm not sure what amp the alternator is supposed to be being a 350 5.7l but it looks quite small for the motor I have had one before it was larger to maintain the battery charge. Even with a full battery right off the charger it still won't fire though. The battery came out of my other 1998 ford f150 and will fire that right up with the 4.6 Triton

Ok, then battery is good, but will start with jump. That could be grounds, when you jump it whether you put the ground cable to battery or frame/chassis and then starts, the two vehicles share a ground. Without the jump, the other end of the battery ground may be not making good contact on the block, or wherever it does attach. Have a look there, follow battery ground cable to where it goes, and take it off, clean eye-terminal on cable and bolt. The connection between dissimilar metals over time can cause a poor contact. Or as an experiment, try just putting jumper cables, both of them from Neg- battery post to a good ground in two places and see what happens without jump.

Have you had the battery tested? Also your state could be bad as well, take it off and have it tested.