The hard lump in you armpit is most likely a lymph node. From your description it appears to be a solitary lymph node and autely inflammed. This could be a result of some infection. You have also mentioned about nodes in your neck. As your doctors have also told you, these do not appear to be of serious consequence like tb or lymphoma
. Take a course of antibiotics like augmentin
625 mgm thrice a day for 5 to 7 days and a pain killer. The nodes should subside with this otherwise you may have to go in for more investigations like FNAC or excision.