He totally would! And he would probably also leave sappy handwritten notes all over their place, like stuck on the fridge or inside James’ laptop.
They are mostly about everyday stuff, like “James, my truest love, will you please do the laundry today? T.H.” but sometimes it will be something more romantic like “James, my truest love, remember that I adore you. T.H.”
James will pretend to be all cool and collected about this but secretly he loves it and never throws away the notes, he keeps them in a box or something and rereads them when he misses his husband. (That is, every time Thomas is away for more than 1 day and a half).