Long post ahead.
OP, you're right that most of those online ones are nothing more than scams to get your email. I did tons of those and never got a penny. Local and the university is where it's at. Scour the newspaper, ask your guidance counselor - apply for anything and everything you can get. Do your parents' workplaces offer anything? Places you've worked? I got $1500 from some random scholarship advertised in the local weekly newspaper probably because I was one of very few to apply. Have you applied for FAFSA? Do you have any idea if you will qualify for grants? Have you had a sitdown with your parents to talk about how much (if any) they are willing to contribute? What do you have in savings? Are you graduating toward the top of your class?
If you get the full-ride to MU, as much as I loved ISU, I can't in good conscience tell you to walk away from that. I am not overly familiar with the mechanical engineering world, but as long as MU's program is acceptable, it'll be your performance and experiences that open the doors. I've done both full-time and intern hiring and as long as I don't go "I didn't even now that school had X program", the school they went to is secondary to the rest of the resume. And I'm hiring for a field that does require a specific group of majors. Now, if MU's program doesn't fall into the decent category, that makes this harder.
Hitting the post-college world with a doable amount of student loans is a huge step up in life. I'd guess with that $22k from ISU you are still looking at what, $80k to fund?
So, a few things then. You need to work with your guidance counselor and ISU to find where more funding may be available. What are the possibilities of going to MU for 1 or 2 years and then transferring to ISU? What I would really hate to see, is you get 1.5 years in at ISU and realize you don't like your major at all and want to change to something entirely different. If you are at all interested in working in Iowa for a year to get in-state tuition, do it.
If you go to ISU, do you have any credits to transfer? How feasible would summer classes (either at ISU or locally - make sure they would transfer if done locally!!) be to help you be done in 3.5 years or at least you give you a freer class schedule that allows you to work more or participate in major-related activities? Beyond overall ISU scholarships, how much and how many does the College of Engineering and Dept. of MechE offer? What kind of students get those scholarships? What % of the students get a paid internship during their time there? What % are graduated in four years? What is the current average debt one of that program's grads come out with? What is the job placement and average starting salary?
And then how does that compare with MU? If the last one, job placement and starting salary (adjusted for location) are dramatically different....that becomes key.
It was expensive to be an out of state student when my husband and I went ~10+ years ago. I can only imagine how much it's gone up. The difference from you is that our in-state option was also pricey and very low on available funding. He and I went slightly different routes. He had built up decent savings through working in high school and had jobs that paid more in college - and were related to his major (CprE). He got some in scholarships but not a ton. His internships paid much more than mine did. His parents also helped out more than mine were able and he came out with no loans.
I got a boatload in scholarship and grant money. I worked and took summer classes that gave me more time for resume-padding activities/work during the school year which led to more scholarships. My paid internships didn't pay as well (Ag) but they got me great experiences which led to more $$$ in and after school. I came out with a manageable amount in loans and had that paid within 3 years.
So, it's probably not impossible for you to make it work, but you need to have a fairly clear route for getting there. I would love for you to get to go to ISU but I won't tell you to just do it and say the money will work itself out. Good luck.