Sherri, as a graduate with a B.S. in ID, I'm surprised you can find schools that are providing associates in the same field. I would say that no, an associates would not provide you the same schooling OR job opportunities as a full degree, as there is a LOT to learn to become a competent designer. I don't know where you're located, but there are decent schools for it in most places in the country. Auburn in AL, Art Center (and others) in CA, UCincinatti has a full degree, Wentworth in Boston, Pratt in NYC... I do not believe that a degree from ITT Tech would make you look good to most employers, but what you do with a degree is your own. I graduated with people who work in law now, and that's that. Most of us have design jobs, and some of us are Design Engineers.
I would also look for schools that provide a B.S. as opposed to a B.A. While it might not make the biggest difference in the long run, B.S. majors tend to learn more of the technical and manufacturing side of the discipline, whereas B.A. majors tend to focus on form and visuals - again, not a bad thing, but a choice. And with the job market like it is now, it is a choice worth noting.
A degree in Industrial Architecture, while similar to ID, is going to be limiting in that Architects learn a different skill set than designers, and usually their job uses are different (buildings vs. objects). Also, if going into Arch, a normal Architecture degree is probably the best bet. Mind you, the job market there is more highly saturated than in ID. Most people who go into DE get a degree in Mech Eng, Manufacturing Eng, Industrial Design, or sometimes "Industrial Design Engineering" which seems to be a rather rare program to find, but perfectly suited for either ID or DE, with more of an engineering bent than most regular ID programs.
Again it is all down to the individual... While I graduated from what is technically a Tech school, with a B.S. in ID, I am one of the few people who I graduated with that cared about math, and that is why I was able to carry the engineering side of design farther into my personal career. It also depends on the business - some people looking for a DE just need an industrial designer. Sometimes people hire industrial designers, when they really need someone with a mechanical engineering background. It is all about keeping your skillset sharp and broad, and finding what you're good at and doing it.