B737-800 Flight Attendant reported a fume event that occurred inflight and the Captain diverted to a nearby airport. The flight crew were then assessed and tested due to the presence of physiological symptoms.

Narrative

As we started taxiing in ZZZZ I was the only one in the back galley getting my safety demonstration equipment and my FA #4 was getting into position for the safety demonstration overwing. I noticed the odor was like a plastic/rubber smell but it went away as the pilots started the engines. We performed the safety demonstration and when I came back I noticed the smell. As we sat down for takeoff my FA #4 asked if she could turn off the coffee pot warmers and I told her yes." At that point I figured the smell came from the coffee pot warmers as sometimes coffee spills on that metal hot plate. As we were getting ready for the service I noticed the smell but since we had turned on the warmers and started brewing the coffee didn't think too much of it. We did the service and noticed it as we got back to the galley. After we picked up the trash I sat down and ate. After like 5 minutes my FA #4 comes to me and asks if I can smell a rubbery/plastic smell. I told her I did and at that point it was stronger. She went up and told the FA #1 so at that point FA #1 came to the back to assess. She smelled it 3 to 4 rows from the back galley; it got stronger as she went into the back galley; and she saw a haze in the galley. At that point she informed the Captain and he advised us to turn everything off and wait 10 minutes to see if the odor subsided and to keep him informed.At that point I became used to the odor and could not smell it anymore but my FA #1 was still smelling it. The FA #3 came to the back and she noticed the odor 3 to 4 rows from the back galley as well. At this point the Captain decided we were going to make a rapid descent and do a [priority handling] landing into ZZZ with emergency vehicles meeting the aircraft as a precaution he did not want to take any chances in case there was something else going on that we could not see. We made the [priority handling] landing; ZZZ emergency vehicles assessed the aircraft using heat detection equipment; and they proceeded to follow us to the terminal. The odor did dissipate as we got on the ground. About 45 minutes before landing I started to get a headache rating a 6 from a scale of 1 to 10 (1 being mild and 10 being severe); very dry eyes; and exhausted/tired feeling. The Captain told us this was going to be written as a fume/odor event; Maintenance came and talked to each one of us; then a ZZZ Manager talked to us. He asked how we're feeling - 3 of us had headaches; red eyes; and exhaustion. He advised us we could go straight to our layover hotel or we can go to the hospital to get blood work done for carbon monoxide poisoning. When we decided we were going to the hospital to get tested whoever he was talking to on the other side was not happy with the response we gave him it almost seems like they were disregarding the entire fume event. We proceeded to the hospital via rideshare after being on the phone for 30 minutes just to file a report and start the process.Fume events NEED to be handled differently and faster. This whole calling to file a report to talk to a nurse to assess AND then we choose whether or not we want to go to the hospital based on nurse recommendations should NOT be handled like this! So much time wasted ESPECIALLY when we have a 4-hour window to get blood drawn! There needs to be a faster way to get this done. Furthermore the attitudes from managers is NOT appreciated especially when we tell them what our symptoms are! In addition; having to call out sick; then file a report; and then get it recategorized should not be the case; it almost seems like we are getting penalized for even reporting a fume event. The fact that if we had worse symptoms; having to worry if we would be pay protected for the trip since we were working and having to a get "sick" before we file to get it recategorized SHOULD NOT have to be this way at all; the airlines' priority should be for us not to worry about that and make a seamless process. Fix this process and make it BLACKAND WHITE!"