On WWWJDIC, there are two entries.
早々(P); 早早 【そうそう】 (n-suf,n-adv) (1) as soon as...; just after...; immediately after...; (adv) (2) (usu. as 早々に) hurriedly; in haste; quickly; promptly; early; (P)
早々(P); 早早 【はやばや】 (adv,adv-to) early; quickly; promptly; (P)
As you can see from the entries above, the parts of speech for each of these two readings are different.
Based on this part of speech info, this seems right to me:
早々【そうそう】にごへんじいただきありがとうございます。 Thank you for your prompt reply.
新年早々【しんねん・そうそう】、借金を催促されるなんてひどすぎる。 It’s too cruel to demand money right after the start of the new year.
This is because these both seem to be nounish usages.