If I am going to do management research on cloud computing adoption challenges and how difference organizations are dealing with those challenges, what research methods I should use for better results?
I suggest looking at case studies and surveys of cloud issues. For example, here is a really early case study (2009) of cloud adoption: Article Developing an Enterprise Cloud Computing Strategy
Security has been an issue with deployment, and here is another study of the issues from a few years ago (2012): Article Some Key Cloud Security Considerations
Contrast that with a more recent case study (2017) on cloud security issues: https://www.intel.com/content/www/us/en/it-management/intel-it-best-practices/boosting-iaas-and-paas-security-in-the-public-cloud-paper.html
You might also want to look at RightScale's 7th annual cloud survey of Cloud users. Here is the latest one: https://www.rightscale.com/blog/cloud-industry-insights/cloud-computing-trends-2018-state-cloud-survey
Would be interesting to see how concerns changed over the years. Hope that helps!