Welcome, don. Thanks for a nice, reflective question.
The cogito has much less importance for Descartes than is commonly supposed. Its significance is twofold: (1) it is (merely) the first truth which Descartes has come upon which (as he supposes) 'is necessarily true whenever it is put forward by me or conceived in my mind' (Med. II: Cottingham, II: 17). (2) Descartes realises that there is very little he can do with the cogito as an isolated truth. What he takes from it is that it is a 'clear and distinct idea' and that 'I now seem able to lay it down as a general rule that whatever I perceive very clearly and distinctly is true' (Med. III: Cottingham, II: 24). Med. III is packed with other ideas that Descartes incorporates into his argument on the ground that they are equally clear and distinct - e.g. 'there must be at least as much in the efficient and total cause as is in the effect', 'the objective being of an idea cannot be produced by potential being', &c.(Cottingham: 28. 32).
The cogito is also not self-standing. Descartes needs to make certain that clear and distinct ideas really are necessarily true. This leads him into what many take to be the 'Cartesian circle'. He proves that clear and distinct ideas are necessarily true by using clear and distinct ideas in order to prove the existence of a perfect God who is no deceiver and guarantees the truth of clear and distinct ideas. (This is a contested area but whatever the case about the Cartesian circle, Descartes very evidently does believe that he needs to secure the necessary truth of clear and dsitinct ideas. So even the cogito, whatever Descartes epistemological relief at its discovery, needs reinforcement.
That aside, and at the heart of your question, Descartes has no difficulty in accepting, 'I feel pain, therefore I exist'. In fact he accommodates just such a case in Med.III: 'But what then am I? A thing that thinks. What is that? A thing that doubts, understands, affirms, denies, is wiling, is unwilling, and also imagines and has sensory perceptions [et sentiens]' (Cottingham: 19). Among sensory perceptions are pains.
As for, **'Then why can't it happen that in dream, I "think" that I think, but in reality I don't', I suggest that we already have the answer to this. What would replace your 'I "think"' except 'I imagine' - I don't think but imagine that I do. But Descartes has already included imagination in thought, therefore in dreams I imagine = think that I think: so I think.
The Philosophical Writings of Descartes, II, ed. J. Cottingham et al., Cambridge: Cambridge University Press, 1984/ 2008.