I have one of each. Long wool coats protect your bum and legs from the wind, but have to be drycleaned, brushed of hair or lint and protected against moths. Puffers are more practical and portable but cover less of you, and depending on filling may also have to be dry cleaned (in the case of down).

The trench coat I have is op shop and the puffer is Kmart so getting both came in well under budget.

If you really can't choose you could start with a cheaper puffer to get an idea of it while you continue saving. Then if you don't like it you can donate it and buy something more suitable

Edit: Also consider where you'd be wearing it most. Woolen coats are generally more formal and puffers more casual.

Thermals are also great to layer under clothing if they're not a sensory issue for you

Edit 2: Also definitely consider adding scarf, gloves, woollen socks and maybe a beanie. They don’t have to be expensive or brand name, just having them helps a lot