I would think it would depend on the age of the child, their language skills, and the support you have from family and friends (not just emotional, but practical bring-you-dinner kind of support). Also, will you have personal leave you can take if you need it? Will the boss support you if you need more time later?
Assuming you are adopting an older infant or toddler, I would recommend you take at least 8 weeks immediately and reserve the 4 remaining weeks for as needed - Maybe you can ask to go back part-time for 8 more weeks afterward, depends how flexible your boss is. It would depend too on how supportive your boss is about you taking leave later. While you might be entitled to it, s/he may be less happy with the uncertaintly of taking it later. You are not required to take all 12 weeks at once, but just within one year of the adoption/birth.
This assumes of course that you can't afford to take all 12 weeks. I imagine that would be optimal. And are you sure you aren't entitled to any paid leave. At my company, the paid leave is only diability or medical leave, so it is similar to wha tyou describe. However, if your company offers "maternity leave" (not necessary medical) you might be entitled ot it.
And last, keep in mind that if you request 12 weeks, you can probably go back earlier. Conversely, if you request less than 12 weeks, you can proabably extend it. Depends what you think would go over best with your boss. But at least you should be able legally, with some warning, see how it goes before you make your final decision. Those 12 weeks are yours and within reason you can't lose your job if you extend or shorten your request. Chek with HR to be sure.
Congratualations, good luck.