+1 with the above.
After you have let your tongue and mouth take a rest, try doing a few things differently.
Let your tobacco dry out for 15 minutes or longer before packing, if it feels damp. I find that I get the most flavor when my tobacco is dry to the touch while still pliant.
Pack your pipe lightly - fill to the top and pack down until about 1/2 way to the top, then fill again and pack that down to about 3/4, then add once more and pack sllightly.
LIght up and get a good coating of charred tobacco over the top, tamp that down and then light up again.
Puff to get the pipe lit, then slowly sip, like sipping a scotch, easing the smoke into your mouth without inhaling, and out through your nose if you can.
You will be surprised what flavors are revealed through a very slow sip. Think wisps of smoke, not wreaths of smoke.
Wait a few seconds and take another sip.
Repeat.
If the pipe goes out, just relight. Keeping the pipe going isn't the point. Getting flavors and enjoying your smoke is the point.
Slow sipping will avoid tongue bite and deliver the flavors.