First, let me say I've applied to several so I know how time consuming it is and what is involved.
My advice is to apply to as many as you want, but realize that it takes a lot of work to fill out all these applications and prepare the information for your recommenders. I would say 6-10 is enough to reasonably cover all your options. Maybe 2-3 schools you know you have a great shot at, 3-4 good schools you have a decent but not great shot at, and 2 schools you'd love to get into that you'd have a slim chance of getting into.
Also, as far as I know there is no engineering GRE(there used to be) so I'm assuming your talking about the General GRE and about the verbal section. I don't think a low score on the verbal will really hinder your chances too much provided you do well on the quantative and have good grades and hopefully good recommendations. What I mean is, it isn't the only factor, so I don't think it's the end of the world if you do bad. You should do well on the quantative portion, and if you are unprepared you can get prepared and do very well. I think it's harder to prepare for the verbal section, it takes a lot of time to learn so many words, but any preparation is better than none, so just do your best.
As for how to get so many recommendations. Prepare a packet for each recommender with self-addressed envelopes with postage already included and explicit instructions, deadlines, and so forth. Make it as easy as possible for them to send the letters. Also, do you really want to go to all those schools? I mean, have you looked really closely at each programs details? I also started off with a huge list but I found my choices getting slimmer as I did more research.
Goodluck.