You need to consult with an attorney, first and foremost! No, they won't work for nothing, nor should they.
Fed Law makes it a Fed Felony to purchase a handgun in any state where you are NOT a resident. Spending weekends in NH will never qualify you as a resident. You can do an Advanced Search here and turn up more on this matter.
That's why you were told that the handgun would have to be shipped to a MA FFL and then you could keep it in MA or in NH. HOWEVER, you would need a MA LTC first, which you aren't going to get!
MGL Ch. 140 S. 131 (d) (i) (e) LTC Statutory Disqualifications
(i) has, in any state or federal jurisdiction, been convicted . . . for the commission of:
(e) a violation of any law regulating the use, possession or sale of controlled substances as defined in section 1 of chapter 94C
BTW, although at some point you MIGHT be qualified for a FID card statutorily, under Fed Law (and because of MGLs) you are FOREVER considered a "forbidden person" to possess ANY GUNS in MA at any time. Fed Law only allows for "full restoration of rights" and MGL only allows for "partial restoration of rights" (FID only, not LTC) and thus you would be facing a Fed Felony if ever found in MA with a gun of any type. IANAL, so check this out with a competent firearms attorney.
So, short of permanently moving to NH, AND depending on their laws wrt "disqualifiers", you may or may not ever be able to possess a gun. I don't think that your phrase of "wiping from your record" for three separate convictions will cut it in MA, but that is where an attorney could help, if anyone can.
Good luck, sounds like you might need it.