Is it obligatory to continue praying Istikhara throughout the engagement period, or is it only prayed when one embarks on a matter? And do the current problems between the engaged couple have a bearing on the Istikhara?

Seeking guidance (istikharah) is recommended in permissible worldly matters. It applies to actions that are neither obligatory nor recommended. It is also permissible in the case of a choosable obligation (wajib mukhayyar), and in matters that are open-ended, such as Hajj. It is forbidden for disliked (makrooh) or forbidden (haram) actions. It is permissible to repeat it before embarking on a matter. Istikharah includes accepting a suitor. Problems after acceptance do not mean that istikharah was not performed; rather, they should be dealt with wisely. The suitor must be treated as a non-mahram until the legal contract is concluded.