clogged pipe or trap issue?
You should have an 1/8 to a 1/4 bubble pitch from the outlet of your trap to the inlet at the "TEE", not from the bottom of your trap. You don't say what type of pipe you have for your drain line, galvanized or schedule 40 PVC, or castiron. You also don't mention if you have vented the laundry drain. "Wet" venting your drain is not a good idea for exactly the reason you mention about the drain backing up. Whether you are using PVC, or steel, or castiron your TEE should have a sweeping curve that runs from the branch inlet around to one end of the TEE and that end should be looking down towards the floor. If the TEE is correctly in place and the pipe has the correct pitch and vented properly both the kitchen and laundry will have no problem draining. Unless you have a partially plugged sewer line under the floor.