Usually heat sensitivity is related to a non-vital nerve. Which means that the nerve is dead and root canal therapy is necessary. I would mention to the doctor that the sensitivity is coming from a tooth closer to the front of your mouth.
get an opinion from an endodontist to be sure.
typically, heat sensitivity over cold sensitivity in a tooth indicates nerve damage and a root canal treatment (rct) would be the treatment in that case.
cold sensitivity could indicate either a temporarily aggravated nerve which may settle over one day to 1 month, or crack which may need a crown and/or rc.
pain worsening after a bite typically indicates a fracture, and these do not always show on x-ray.
although....sometimes even the professional may be in error. or their philosophy may be that rct may be best now, because later, we see patients come back for it anyway.
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my story, very short:
i cracked my tooth (molar #15) on grain of sand. my tooth was sensitive to cold and the slightest touch. there was great pain upon biting, with relief after a few seconds.
after six painful months arguing with three dentists/endodontists against rct, i was finally prepped for a gold crown. i agreed to proceed with rct *only* if pain persisted after receiving temporary crown.
at the present time, one year plus later, i have no pain at all - with only a crown!
actually, i've got the same thing happpening on the other side now. :(
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bottom line, consult with an endodontist. maybe even two.
remember, you might need some more time to let things heal, although the heat sensitivity may be a concern.
i have heard of "crossed nerves" where pain is reflected to different areas. have you checked with your dentist about this?
best of luck.