Happy to report that I bought two separate sets of train tickets on SJ's website last night. Looks like they changed payment clearinghouses, which connects to Visa and asks to call/text the phone number registered with the card with a code that you enter on the website. Also purchased tickets on Bus4You's website without issue (but the first time always seems to work--subsequent times are usually iffy).
I still need to purchase one set of SJ tickets, Arlanda Express, plus Flygbussarna tickets, which I will do when I arrive in Sweden. Wish me luck.