Я поправляюсь ... is that correct? :)
Almost so. I think an average native speaker will be somewhat surprised but will do get the desired meaning. Usually for an
animate subject «поправляться» means
• to correct oneself (after a mistake etc.);
• to be recovering from an illness;
• to put on weight [for which there is a somewhat less neutral variant «толстеть» which almost literally translates as «to become more thick/fat», but some people don’t like that be said about them or someone else].
For «get better» used with an animate subject there’s a more preferred translations such as «
исправляться» (yeah, subtleties of verbal prefix usage in Russian are a mess alike English phrasal verb preposition ones) or «улучшаться» (almost the same as «становиться лучше», almost literal translation of «get better») or something. Although, in this case I think a more specific phrase such as <oops, can’t conjure anything good at the moment; I hope someone will help me!> seems more natural.
There are discussions about this stuff on russian.stackexchange and Tatoeba forum but sporadically they are misleading (after all it’s the internets), e. g. natives and non-natives alike often tell nonexistent things about phonetics and phonology.