"I've" is not the contraction for "I would" (or "I would have"). "I'd have" or "I would've" are both correct contractions. "I've" is a contraction for "I have", and so your example sentence says:
Once I have been able to get over this [something], I feel more like myself (as in now).
With a future condition, either the future simple or the future hypothetical are OK:
Once I am able to get over this cold, I will feel more like myself.
If I am able to get over this cold, I would feel more like myself.
So I'm not quite sure I understand your question. But if you are asking whether "would+infinitve" in one part of a sentence can be paired with the simple present in another, then yes:
I would give you a hundred dollars if you eat that bug.
The example you give from the book is set in the past, but you can say something similar in the present tense:
Once you put the pieces back together, you are never quite the same as you'd been before the fall.
Once you put the pieces back together, you will never be quite the same as you'd been before the fall.
If you want to refine your question I can give you a better answer.