Try increasing Fidelity to 100% in Envelope Options, that worked for me!

I was having the same issue. The answer that starts with a rectangle worked for me:

1. Draw a straight box
2. Envelope distort that with your outlined text
3. Then warp that by adding points and creating curves

This provided much more control and better results.

I´m having this problem too. I´ve been using

object > envelope distort > make with top object

and it´s been working the way I want (red text) to until now, but now its suddenly making it all wavy, see selected text. How do I fix this?

Envelope the text into a square/rectangular box. Add points in the center or areas you want to change and then move from there. For instance, to make text that is flat across the top and then the bottom rounds up toward the middle, you would add a point on the center bottom, use the direct select arrow, and move that point up to round it out. You may have to also move the handles on other points but this may help solve your problem. It solved mine.

I had this same problem with a few outlined letters in my design. Although I had deselected "Distort Appearance," a few letters were distorting when I used Envelope Distort > Make with Warp...

I was able to solve the problem by selecting all of the points in each troublesome letter (these were outlined and therefore closed shapes) that was distorting and using the Unite function in Pathfinder. This seemed to clean up overly complicated handles and curves in my letters without changing the look of them and when I tried Make with Warp... again, it worked perfectly!

I hope this helps.

I am having this issue as well. The shape I am using is pretty straight forward (is asymmetrical). So is the font I'm using (simple sans serif).

If the shape you are creating for the top object is not symmetrical it is going to skew or twist the type. You will not that the tutorials always choose symmetrical shapes.

Half the time I get curving stroke distortion from a straight stroke font. The top (a horizontal line) comes out fine but on the bottom (a slightly arced line) the letterforms are squeezed to either side. The horizontal strokes are distorted. Both sides are vertical. Tried with both top and bottom arced and it seems to work, but that is not what the client needs.

Having the same issue as you and Amanda. Driving me crazy.

I am having the same issues. The envelope distort is adding anchor points to the object AFTER I click "make with top object" - even if I'd made the shape with straight lines using the pen tool. Even worse, the added anchor points have handles and can't be converted using the "convert anchor point" tool.

Please show pictures.
You are doing everything right, but it all depends on shape of envelope path, it's complexity, number of points, handles (especially handles). When you make distortion 'with top object', it got CONVERTED to mesh automatically, so you have to understand meshes anyway.
I would like to help, but I need real art you making to give you an advice.
Also please look for free MeshTormentor plugin which is must-have for working with meshes.