One label prints, then red light comes on. We

Most people I know print addresses from a sheet of labels, which ejects at the end. It sounds like you are printing one address from a continuous roll of labels, and they are so close together that the printer cannot push it out for you without misaligning the next one. I think both systems are designed mainly to print a batch of labels at one time.
If you are printing a batch, it is best to prepare the whole set first and save it as 'data' for printing addresses. For a one off or just a few I would use a sheet of 24 labels (say) with 3 to a row, and 8 rows.
Does this help?

have you checked the size of the label in your driver?
You may be simply sending the information on two pages!

Did you try to reload the labels. Pull the labels out and clean the paper path

This is an easy printer to fix since there is only one control button. Turn the printer off. Hold the green feed key down while turning the printer on; once the red light begins flashing, let go of the feed button. This will do three things: the printer will calibrate with the installed media, it will print a configuration label and the printer will go into dump mode. Press the feed key again. One more label will come out with with the words "Out of Dump" printed at the top. If the printer still fails to print, remove the ribbon and try the same procedure with direct thermal labels (UPS or FedEx). From there, consult the manual for more troubleshooting tips.

Your physical labels size is not matching the label size set in your printer driver and also you need to carry out a callibration.

Turn off the printer
Hold down the front button
Turn on the printer
After the light flashes 2-3 times red release the button
The printer will now callibrate
After callibration press the button once

Have a look at the way the driver is setup, by sound of it you might have it set for thermal transfer, it needs to be set to direct thermal. Thermal Transfer is for the TLP2844. the next time you print will send the changes to the printer

Recalibrate the printer and you should be good to go. Do this by turning the printer off, press and hold the button and turn on the power. When the light flases red let go and it will calibrate, at the end it will do a printout. Reboot the printer and your fixed.

call the customer service.

1: Is it a TLP2844 or a TLP2844Z.

Operationally they are completely different printers. TLP2844 uses ELP (Eltron Programming Language. but TLP2844Z uses ZPL (Zebra Programming Language). You need to calibrate the printer to the labels.

Calibration:
If its a TLP2844. Load media. Connect power.
Hold down feed key, power on, count one, two, release feed key. The printer will go through a calibtation routine and print config label. IMPORTANT - When it prints this label you must press the feed key and see the message OUT OF DUMP. Your printer will now be calibrated. Check this by hitting the feed key a few times and printer will feed one label each time feed is hit.

Calibration:
If its a TLP2844z.

Load media. Turn printer on and wait for green light (Approx 1 minute)
Press feed key and keep it pressed until green light blinks once, then twice, now release feed key.

Check the tension on the Ribbon Take-up Spindle there is a nut on it that sometimes can loosen or tighten its self and will throw off the lables. Its a nut that has a of couple Hex nuts on the corner, screw the nut to about the 3rd thread from the top of the bolt facing you then tighten the hex nuts on the corners of the nut. Then try printing the labels make sure the media guide also is set to the lable size that is installed.