seriously, every other recruiter asks me if I have green card....should I just say yes and later show them I have TPS?

Just say you're eligible to work and have valid work permit. If they ask for status provide TPS information. It is against the law to discriminate on the basis of citizenship, so you shouldn't have any issue applying for the job.

when applying for jobs, if the application asks you if you are green card holder ? Do you say yes ? I feel like I have lost few jobs cuz I tell them I have tps and they just don’t know about that

No, the recruiters generally do not care as long as you are permitted to work and the duration is long enough.

My two cents: Never say you need sponsorship in the application. There are chances of getting filtered. When they ask about Green Card, you can say I have work authorization. I am permitted to work. Bring up sponsorship only after there is a job offer. Yes, they might not move forward but bringing it up after the offer becomes a part of your negotiation.

what do you say when they email you and ask are you a green card holder or us citizen? this is only for GC or us citizen?.....and also when filling application when they ask are you a green card holder? do you select "yes" or no in this case

@Shreeram they have emailed me and ask are you a green card holder or citizen? this is specifically for GC and citizen only? what do you reply in that case

Just be honest in that case. Mention that you have Temporary Protective Status which can be renewed every 18 months and you are authorized to work in the USA. If it doesn’t work then you’ll find something else. Try not to lie as much as possible regarding job application. They might as you to provide GC if you say that. Also, you should find work where they will sponsor you. All the best!