It'd difficult to make exact rules for your example because the significant words -- feel, seem, and bad -- all have a wide range of meanings which vary with context.
While seem usually refers to external impressions, feel usually refers to internal impressions:
She feels tired (= she believes herself to be tired)
She seems tired (= from looking at her I believe she is tired)
However, "feel" can sometimes refer to external impressions. In this case both feel and seem mean the same thing. For example:
This alley feels/seems dangerous. Let's go around. (= from looking at it, I get the impression the alley is dangerous)
The weather feels/seems nice today. (= My impression is that the weather is nice)
Lastly, bad has a very wide range of meanings, from "ill" to "wicked" to "dangerous" to (in slang) "perversely good". This does make certain sentences ambiguous:
He feels bad.
Without further context, I don't know for sure if this means, "He is having sensations that make him believe he is sick or upset," or "Looking at him, my impression is that he is a wicked person." Usually you can judge this from context:
Martha felt bad about not telling the truth, but she had promised to keep George's secrte.
Martha watched the strange man slink down the street and vanish into a narrow doorway. He felt bad to her, like a ferret stalking some hapless mouse.
Remember, ambiguity is often common and natural in any language. Sometimes this is done purposefully to give a sentence multiple meanings, and other times the actual meaning should be obvious from context.
(Edit) Idiomatic use of these verbs requires experience and practice. For example, "The food feels delicious," is not idiomatic, since "to feel" generally refers to sensations that relate to touch. Even in the context of something like:
It feels like we're walking into a trap. Let's get out here.
"feels" relates to the prickling sensation on your skin you get when something seems wrong in your environment, and it makes you fearful or suspicious. If you want to refer to the other senses, use verbs like smells, tastes, sounds, looks etc., "The food looks delicious".
Also, in most cases where there is some ambiguity, one interpretation is generally much more likely than any other interpretation. Of course, this ambiguity can also be used for humorous effect, when one interpretation is assumed while another is meant, as in the most recent Star Wars movie when Luke tells Rey to "reach out" and "feel" the Force.