For the first example, you can move nature to the front of the sentence like this:
I got hung up because I wanted to understand the solution before starting. Nature had other ideas.
I got hung up because I wanted to understand the solution before starting. Nature doesn’t work that way.
Alternatively, you could re-work the whole phrase into one sentence.
I got hung up because I forgot that nature brings understanding with doing, not before you’ve begun.
Nature reminded me that trying to understand the solution before I started the process wasn’t how things work.
There is value sometimes in short sentences, but it also needs to be balanced with providing information and quality writing. I don’t think it’s a problem for a sentence to be a little bit longer. Even in online writing, our guidelines usually said 15 words for advertising type writing, and those are only 17.
For the second example, I would rework the whole sentence.
In exchange for a basic income, you provide the care for another person twice a week.
Accepting responsibility to care for another person twice a week provides a basic income.
The way it’s written in the original form is a little bit awkward, if you were having a conversation with someone, you wouldn’t likely phrase it that way. It sounds stilted instead of flowing.