Unfortunatelly I have similar issue, and I cannot fix it neither by opening and saveing, nor by bublishing and unpublishing.
My problem is:
Whenever I change payment method, flow advances to the next page. It works OK with delivery method, but not with payment method.
Moreover, and whats worse, when I choose 'Płatność przy odbiorze' what means 'Payment on delivery', and I advance (or it advances automatically) to another page where I register and I click 'Dalej' what means 'Next', then the information about selected payment method is lost and the system behaves as if default option was choosen.
Page:
www.manufakturakosmetyczna.pl/
Joomla! 2.5.9
HikaShop 2.1.3 Business