Hi,
First, please go to in the "advanced / legacy" settings and deactivate the "bootstrap design".
Because that legacy setting is still activate, some part of the interface are still using "bootstrap 2 style" while your template does not support it.
Afterwards, you could see to try the new checkout instead of the legacy one. That new checkout do not use anymore the "bootstrap 2" styles for the selection of the shipping or payment.
Otherwise, you have to fix your CSS. If you were using HikaShop 2.6.4 and it was fine, it means that you had working CSS for that ; and with HikaShop 3 and the legacy mode, the content is still the same ; so you should not have these issues with the updates.
You must have change elements in your website/template which make that.
Regards,