You may be having a crank postition sensor (CPS) issue. A lot of Avenger/Sebrings have this issue at some point
to replace, remove the batter, and battery tray...and down underneath the tranny mount you see a little black cylinder sticking into the tranny, held by one 10mm bolt, thats the CPS
Directly below the distributor.
..and it looks like this..
Be careful of how it is installed though. The paper spacer is very important and it can determine whether installation is successful or not.
Price for the CPS ranges anywhere from ~$50-$100+, all depending on where you get it. You just have to shop around the internet and local autoparts stores.
Please check everything else first...spark, fuel, power. Check your fuses, relays, and connections. On the firewall, there are a set of relays that can also affect start or no start. Your fuel pump relay and ASD relay are on that set of relays. I would suggest getting a Haynes/Chiltons manual also.
Turn your key to the ACC position...can you hear the fuel pump prime? How hot is it where you are? is your thermostat getting hot looking at the gauge?
Do your normal quick and easy checks for fuel and spark. I'll post more when I think of more and/or find more info for you, but we'll start there.
...and welcome to DF.