Traditional: 舍命 vs. 捨命?
Some general principles:


In early Chinese writing, there weren't enough characters to represent the range of words or morphemes used across the language;
Many characters became used for multiple purposes, and some of these purposes are not etymologically related to the original (phonetic loans);
Since multi-use (including phonetic loan) characters can become very confusing to read, components are piled on to existing characters to distinguish between different senses;
This is why Chinese writing has a large inventory of characters, some of them with many components.


The natural development of Chinese writing is therefore from:


A limited number of characters with simple shapes / few components, to
A large number of characters with complex shapes / many components.


In addition, since Chinese writing has a very well-recorded history, many words come from old literature but survive in modern expressions, and their spelling frequently become fossilised, regardless of whether the characters making up those words have changed (became more complex through the addition of more components).

With the above principles, we should be able to see that this question is in fact the same question as Why 纹 is the wrong character to write for tattoo?.



西周金夨令方尊集成6016

「舍」 is comprised of phonetic 「余」 (Baxter-Sagart OC: /*la/) and a distinguishing mark 「口」, indicating the original meaning to give/donate something, extended to to give something up / abandon something.


  《易・比》：“舍逆取順，失前禽也。”



The original meaning, to give/donate something, is now written as 「予」 (/*laʔ/), which is a phonetic loan character;
The extended meaning, to give something up / abandon something, is now written as 「捨」 (/*l̥Aʔ/), formed by piling on semantic 「手・扌」.




「舍」 itself does not mean either of those things in modern Traditional Chinese, being (almost) exclusively used as a phonetic loan character for the meaning dwelling (/*[l̥]Ak-s/). However, modern spelling standards are pretty arbitrary, and you're unlikely to find that most people stick to them rigidly.