I found in different occasions さま after a verb, but I'm not sure about its meaning. For example, in 大辞林's definition for 半ば:
ある状態に半分ほどなっているさま
（「…てはじめて」の形で）さまざまな経過を経てようやくその状態になるさま
I noticed both times it's a なるさま form, so I'm not sure if it's a fixed form, but I can't find it on Weblio, so maybe not.
Maybe it's meaning 3 on Jisho (state; situation; appearance), but if that's the case I don't really undestand what does it mean in those sentences.
さま (様) just means "state" or "appearance", the same as 状態 or 様子.
See the first definition in デジタル大辞泉:
さま【様／▽方】 の解説
［名］
１ 物事や人のありさま。ようす。状態。「雲のたなびく―が美しい」「物慣れた―に振る舞う」
It's used heavily in dictionaries in this way, and also in formal texts/presentations, but you're not likely to encounter it in everyday life outside of those situations.
As to how to interpret it, borrowing your examples we have:
ある状態に半分ほどなっているさま → The state of having become roughly halfway a certain way
and
さまざまな経過を経てようやくその状態になるさま → The state of finally becoming a certain way after having gone through various things
and for an example without なる (from the definition of 断然):
程度が他から非常にかけ離れているさま → The state of the extent/degree of something being far removed from others.
