Help with simple sentence please

Please can someone help? I'm reading this sentence - "無理せずにあがって来い"

The person speaking is talking to the main character who is standing in freezing cold water trying to clean herself - I get that the second part is "get out [of the water] and come [here], but what is the best way to interpret the first part? I've not been taught せずに, but apparently it's a sort of negative form of suru meaning without doing?

So, "without doing impossible get out and come"?? Is the impossible thing the speaker is referring to washing in freezing cold water? Does that sound right!?

無理する is often used to mean "do something unreasonable/work yourself excessively" (push yourself to wash in freezing cold water, for example). I think you've got the broad gist of the meaning.

Another example of a similar use of せずに would be something like this (遠慮する means to hold back, often out of politeness)
遠慮せずに食べてください
(lit. "without restraining yourself, please eat", more loosely like "feel free to dig in")

Aaah thank you so much! Sometimes figuring out what a sentence means (as opposed to just a direct translation) is so awkward... That really helps!