perm crown and some issues

I have just perm crowns put on tooth 12 and tooth 13.  Both tooth have been in pain for different reasons.  Tooth 12 has gum issue from the previous crown which my gum was allergic to.  The gum is very much itchy and very much irritated the more I eat, and clean, the more it gets irritated.   It is fine as long as there is no food or clean.  my dentist suggested deep cleaning which may help.   my husband and I have not decided yet especially that it is very costly.   do you have any other solution which can be less costly but effective as well?

Tooth 13 does not have root canal.  After the crown, the tooth is terribly sensitive.  At the beginning I thought the sensitivity is coming from the whole tooth.  But, I found out that the sensitivity is coming from the edge of the crown around the gum line on both sides.   my question is:  can those exposed sides be covered?  And, If so, how?  I was also wondering why the crown did not cover the whole tooth including those exposed sides ?  thanks alot in advance.. regards