-(으)ㄹ까 하다, -(으)ㄹ까 보다, -(으)ㄹ까 싶다 all share the idea of wanting to do something, but there is a difference in connotation.
1. -(으)ㄹ까 하다 = expresses a fairly clear intention, although weaker than -려고 해 or -(으)ㄹ 것이다.
2. -(으)ㄹ까 보다 = expresses a more uncertain and tentative intention, or an impulsive thought to do something.
3. -(으)ㄹ까 싶다 = expresses similar feelings as #2, but sounds somewhat calmer.
If you exclude -고 싶다 which describes a clear desire, 싶다 is more widely used to indicate a feeling of suspicion or hunch (너무 힘들다 싶으면 그냥 포기해, 비가 그칠까/그치나 싶더니 다시 온다) than an intention or desire. Of the three above, I think -(을)까 싶다 is the least frequently used expression.
Examples:
- 나 직장 그만둘까 해. (Sounds fairly definite)
- 나 직장 그만둘까 봐. (less certain, and sounds like uneasily asking oneself with misgivings)
- 나 직장 그만둘까 싶어. (similar, but sounds more cautious).