The idea of pulling a credit report to show employability is fairly well discredited. In fact, in this day and age of ID theft, i wouldn't supply a Social Security Number until an offer of employment was on the table. It can be made contingent on passing a background check and so forth that will use the SSN at that point, but there's no reason for them to have it until that point. MY data has been compromised enough times through theft and stupidity at government, hospitals, universities and such that I don't give out practically anything anymore.
A credit report doesn't show work history per se and an employer using it for that is poorly informed. It may show employers and may not, but is usually out of date in my experience.
I too have a resume full of consulting gigs. It shouldn't be any harder for them to verify you worked for those companies than any other job verification. Again, in this age, companies won't provide positive or negative references, just confirmation of employment dates because they'll get sued for any comments that could be construed negatively.
If you're being paid like a consultant, you're not an employee of that company, at least in a legal sense it seems to me. Reflect your status on your resume. If you're working under the table as it were, I don't see a reason to not list those jobs. They still show you were working and gaining experience. You might choose to list a number of recent consecutive past employers under a consulting entry rather than as an employer, indicating your were free-lancing and worked for these companies doing X tasks with Y achievements.