When your fly is getting refused and you suspect drag, work through these adjustments in order. Don't jump to changing the fly until you've made a real effort at each step.
1. Change your angle first.
This is the single most effective adjustment, and it costs nothing. If you're casting directly across the current, move upstream. An across-and-upstream cast puts more of your line above the fly on the water, which means the current has to work harder and travel farther before it can belly the line and pull the fly. The drift starts cleaner and lasts longer before tension develops.
If wading directly upstream isn't possible, try a more downstream angle. A downstream reach cast can put the fly in front of the line, buying you time before the current catches up.
2. Add a reach cast.
The reach cast is a mid-air correction. You complete a normal forward cast, but as the fly is still in the air, you swing the rod tip upstream. This places the line upstream of the fly at the moment it lands. Instead of a straight line pointing from you to the fly, you've got a curved line with the midsection above the target. The current has to eat through all that upstream slack before it can put tension on the fly.
The reach cast extends your drag-free window significantly without requiring you to move positions. It's one of the most useful tools in dry fly fishing and it's worth practicing on slow water until it becomes automatic.
3. Mend immediately after the cast.
If the reach cast wasn't enough, or you weren't able to throw one, mend the line right after it lands. A mend is a flip of the rod tip that repositions the line upstream of the fly without moving the fly itself. This is where most anglers make the common mistake: they mend too late, or they mend too aggressively, and the line movement transfers to the fly.
A good mend is quick and controlled. The rod tip makes a smooth arc upstream, rolling just enough line to reposition it. The fly should barely quiver. If the fly moves noticeably when you mend, you're putting too much force into it.
Also, mend early. The longer you wait, the more tension has already built in the line, and the harder it is to move it without disturbing the fly.
4. Build in stack slack when nothing else is working.
When you're fishing very complex water with multiple current lanes between you and the fish, sometimes a reach cast and a mend still aren't enough. A pile or stack cast adds extra slack right at the leader by stopping the rod tip abruptly at the end of the forward cast, which causes the line and leader to fold back on itself and land in loose coils. As the current pulls that slack out, the fly drifts freely before any tension develops. It's a useful technique for slow, slick water where fish are especially wary.
One more thing: be patient with the drift. The angler who makes one good cast, waits three seconds, and picks up is leaving fish unbothered. Let the drift run out naturally. Make your presentation count before you recast.