You can indeed use "Je sens que..." but it would be more a feeling than an opinion.
"Je sens qu'il va pleuvoir" "I feel that it's going to rain"
"Je sens que je vais être en retard" "I feel that I'm going to be late
As you can see it is mainly used to express a feeling toward something that is going to happen...
On the other side, "Je me sens que..." doesn't exist, but you can use "Je me sens.." :
- malade / fatigué / perdu dans cette ville (I feel like I'm lost in this city)
"Je sens que" has the peculiarity that it's most frequently used for negative things. That is either what you're expecting is formulated with a negative (je sens que ça ne va pas tarder, "I feel like it's coming up soon") or is something that would be bad if it's accurate (je sens qu'il me ment, "I feel he's lying to me").
It's not universal (cf. je sens que je progress, "I feel like I'm making progress), but searchig for "je sens que je/tu/il" etc. is pretty revealing.