Only after completion of varushaptikam, Gaya shraddham must be performed. References on Gaya shraddham are found in Srimad Ramayana, “yeshta vyaa bahavaa putraaH yad ekopi agayaam vrajet”. That is, one must beget several sons in the hope at least one of them will perform Gaya shraddham. Gaya shraddham is important.
Akshaya Vatam is a divine place at Gaya.
It is a common practice that when one goes on a pilgrimage to the North, one also goes on pilgrimage to divine places in the South. In our tradition, we do not go to Rameshwaram. Only Sethu snanam is required at Dhanushkodi, which must certainly be done.
There is no need to complete all of these before going on a pilgrimage. Once varushaptikam is completed, one can go on pilgrimage.
It is alright if your mother is unable to accompany you to these places. As a son you can do the needful.