A Modest Grammatical Proposal

In the past 30 years or so, some people have taken objection to the use of the "generic he" in the English language. The result is that "Everyone needs to get out his book" has become turned into the ungrammatical "Everyone get out their book." The first construction, while grammatically correct, offends some people. Many opt for the second construction, but it is grammatically incorrect, since everyone is singular and their is a plural pronoun. So far, no attempted remedy has met with the satisfaction of everyone in society, so I would like to following modest proposal for fixing this potentially offensive quirk in the English language.
Instead of using simply "he" or simply "she" (or worse, yet, some "/" combination such as he/she or s/he), I propose that we create a new pronoun for use whenever both genders should be included in the pronoun. This pronoun, in its basic form, will be called "Shim" (a combination of she and him). This pronoun would have the following forms:
Subject: Shim (Example: As Mr. Jones talked to each teacher, he found that shim was intelligent, loving, and sanitary.)
Object: Shim (Example: As a teacher, shim found it was hard to have students talking to shim.)
Possessive: Shis (Example: Shim went to the store to go get shis favorite food.)
Reflexive: Shimself (Example: Shim went to the store to get shimself some of shis favorite food.)
Alternative possessive: Shine (Example: Whose food is this? It could be shine!)
With this new construct, hopefully we will finally solve the incessant war between the angry feminists who feel oppressed by standard English grammar and the traditionalists. It may take a while for it to catch one, but I think it has quite a bit of potential. I invite each of my readers to submit shis comments.