In my opinion, there are better soaps out there than Col. Conk; I only used it for a few shaves then gave up on it. Maybe if you use a water softener it's OK, but I've never used one so don't know if the Col. plays nicer with some waters than others. You might try a puck of a nicer soap, something nicer than you can get at the drug store or Wal-Mart. I've tried those brands (Williams, Van de Hagen, Conk) and my experience was that nicer soaps and creams do make for closer & gentler shaves.

About pressure: If other factors are off, the temptation is often to push harder. With a properly sharp straight, you should only try to gently scrape off your shaving lather. The whiskers will come off as a side effect. If it's not sharp enough, you'll get the pulling and if you compensate with pressure you're going to get more pain from razor burn.

You might try a razor from whippeddog.com; it'll be cosmetically challenged but it'll be a shave-ready vintage blade at a great price point. Maybe you've got the sort of beard that really needs a wedge. The benefits of a second razor are 1) you have a known good blade to compare your current one to, 2) you'll have one in house when you send the other out for honing, 2.5) If you're honing your own, it gives you a reference edge (I guess this is the same as benefit 1), 3) you can try a different grind. There are other benefits, but they're less virtuous. Things like priming your RAD.

I hope you find a solution. You're a good man for sticking it out despite the pain.