Thank you all very much for your comments, as well to those who emailed me with their thoughts directly.
In sum, it seems that I will need to collect physical cards from these missing DXs, and submit them to a local checker. Thanks to Brian's link, I have a found an ARRL checker within 50 miles of me; I'll take my cards and a printout from ARRL's online QSL entry form to him when ready.
All of your suggestions were excellent, and I've tried a few of them, one being researching the DX's instructed methods on QRZ.com for requesting a card ($3 via Paypal etc.); some have instructions, some don't. I've also checked for my callsign in some of their ClubLog accounts - again, some upload their QSOs while some don't - it's a real crapshoot. I've sent off a few requests with funds via ClubLog's OQRS, but have not seen anything sent to me for my efforts - one of you mentioned your return ratio as being 80% - looks like I'm hitting the 20% first. I am now considering having JTAlert filter stations with LoTW accounts, I'll consider others on a case by case basis.
In any case, I have my question answered - thank you all, again, very much!!