Dear Adya,
You should consider the item to load in that factor where the loading is greater. In case the difference is greater than 0.30 you may consider the item to load on that factor where the loading is more. However, you can also delete that item where it is cross loading on two factors. As a researcher the call should be yours. Please do not be confused. I suggested you to retain items where the difference is more than 0.30 so that you do not miss on the vital data that you have collected. Therefore, before going for data collection, it is highly advisable to get the content validity established based on literature review and expert opinion.
Again, I suggest you to watch Neeraj Sir's video on EFA on how to fix cutoff values in SPSS. The link is as follows.
Coming to your question of a single item factor, usually a factor is composed of 2 or more items. You can cite some papers where an item is composed of one factor and go ahead with your analysis if you wish to retain the item in your analysis. However, you cannot check Cronbach Alpha reliability of one item factor.
EFA is a dimension reduction technique. It is in no way used to establish validity of the constructs. In order to assess the validity of the constructs (convergent and discriminant validity), one should go for CFA. CFA is also known as the measurement model which is a technique available under Structural Equation Modelling.
I personally advise you to go for CFA as it is a PhD thesis. Who will take your measures seriously if the validity is not established?
You can present the reliability figures of each dimension in EFA. However, post CFA, some items may have to be removed due to lower loadings, then the reliability might change. So when you calculate the discriminant validity using Fornell and Larcker Criterion, you may present your analysis in that table along with Cronbach Alpha reliability, Composite relaibility, AVE, Correlation figures etc.
Please find some papers attached for your reference which will solve most of your query. I request other members as well to share their opinion on the question of Adya.
Regards,
Prof. Arunkumar Dubey