A week without food is nothing to the FBT, so calm down. After four weeks, you can en get excited, IMO.

Does your viv have plenty of water for your FBT to lounge and swim in?

You said you raised the temperature... from what? to what? They like cooler temps so I'd shoot for 68 to 74F and wouldn't worry if it was as low as 50F occasionally but I would worry if I were getting in the 84F and higher.

As for feeding..... just put the crickets in the viv at a regular time every two maybe three days and leave your FBT alone. If you try to coax it to feed you probably are just stressing it.

What about the syringe? Well throw it away, IMO. If your FBT doesn't figure out how to eat, nature will take care of it. Sad, but that's life.