The red dot you see in the instrument cluster is an indication that the immobilizer or theft system is activated. Causes for this can vary a bit, so this is what to look for.
If you happen to have a grey-colored key, it means that your car is equipped with the Sentry Key system, referred to as SKIM by its acronym. These keys have a transponder chip embedded that is read by the car --and if it's recognized-- will allow engine
start. A failed key or failed SKIM module can cause this situation where you have no starter operation, so the easiest thing to try would be a second key if available.
But I'm not certain that the 98 Sebring
was equipped with SKIM so we move forward.
Check the small yellow (20 amp) fuses in your underhood fuse block. Bad starters are known for causing the PCM/ fuel pump/ starter relay fuse to blow, which completely derails the start process. If found to be blown, simply replacing the fuse will likely be all that is needed since you've already replaced the starter.
Lack of communication between SKIM, PCM (engine controller) and/ or BCM (body control module) will activate the red flashing dot in the dash as you've seen. Chances are very good that the fuse issue I mentioned will be the cause because the PCM will not be conscious with the fuse blown.