I glanced over the other responses but didn't find anything like my response, but apologies if I'm repeating here. Also, below are a lot of notes on draw, but I think #4 is what mostly addresses your concern about not coming off straight on your draw... But all of the steps below matter for a good draw shot.
1) Line-up, get near parallel with the table but not perfectly so. Watch a pro-level player draw, they mostly aim slightly downward, just a few degrees... It's really worth watching how they do it. Plus it's hard to get flat anyway because often your stick is half over a rail, which causes some elevation. Too much angle though and you're pushing the ball into the cloth which kills the draw and your shot.
Keep in mind most players don't know what angle they are at and think they are flat when they are at a bad hard angle. So if you can, video your shot or have somebody watch you shoot and tell you how are you are doing. It will take practice to get a sense of proper angle to the table.
2) Don't use a death grip or even a solid grip. Try a loser grip, and some players will grip with their fingers, rather than their hand. When they stroke they let their fingers control the cue, and even let their wrist sort of flick in the last bit of stroke. Finding what works and getting a good feel for it will take practice, but definitely avoid the death grip or even medium grip.
3) Old timers would likely tell you to bridge a little closer than normal to the cue ball when you stroke. This will keep the tip on a low path in case you drop your elbow. When you drop your elbow, the cue tip elevates as it moves forward killing the bottom English (draw). So bridge closer, or don't drop your elbow.
4) Stroke is everything, and I think this is where the big mistake is for most people. Most players stroke trying to push through the ball as far as they can as hard as they can, but that leads to over-stroking which is likely going to be crooked, and the hard hit will not impart as much spin as expected or wanted.
Instead of focusing on the forward motion of the stroke, give as much attention on the backward swing... It should be bigger on the back-swing than normal, and that will improve your stroke speed and follow-though on the front swing automatically. This is what gets you a good draw without hitting hard, that good stroke. When you have a good backward motion, you will find that your natural forward stroke will take care of itself (the tip will land maybe like 3 to 5 inches in front of where the cue ball was sitting.
I described separate pieces here that each need to be considered and practiced, but you use them together as a uniform movement to make a good draw stroke.
I'm sure other will have advise that is different, or even contradictory, if that works great. I collected advise from some very well known sources, so it's not mine but rather pro advice. Also, advice is easy; you need to drill this stuff and practice it til you perfect it.
Good luck.