How can you get people to stop shortening transgender and transsexual to the word trans and lumping them together?

The way to stop this is to protest in any way you can. Refuse to do business with those who lump Mainstream persons who were born with transsexualism in men who choose to become transgendered. Contact the news media and let them know the difference in the two terms. If you have TSism or have corrected it with surgery and are now cisgendered, then be sure to object when someone calls you a "transwoman."

Genuine transsexual people are part of the Mainstream, cisgender Community, while transgender people are a subset of the LGBT Community. Saying that transsexual persons are a part of the LGBT or TG communities only harms them and prevents them from being able to assimilate as every genuine TS needs. A number of transsexual persons oppose gayness and recreational transgenderism, and nearly ALL TSs considered themselves a part of the non-LGBT community and not any TG nor LGBT community. The whole point of transition and surgery are to assimilate, not become stuck in a half and half community.

Transsexual persons were well on there way to achieving full rights until male fetishists choosing to constantly dress in drag infiltrated and took over the transsexual community. The problems transsexual persons face are inborn, while most problems the transgendered community face are unfortunately self-inflicted. A person born in the wrong body must live as a member of the correct gender, while it is merely optional for the person who chooses transgenderism.

You can't stop people from shortening transgender and transsexual into trans. There's no reason to do this either, transgender and transsexual people are part of the same community and face the same issues.