I know that 'owe' is a stative verb, but recently I have also seen it in the present continuous.Is there a difference?
For instance :
I'm owing 3 months rent.(shouldn't it be I owe 3 months rent)
I don't recommend that learners use 'owe' in this way, but you will cetainly hear it. I have noticed a tendency in recent years for some of those verbs traditionally regarded as stative to be used in progressive forms.
Don't forget that the 'stative/dynamic verb' distinction was always a little artificial. Some writers preferred to say something like 'verbs normally used statively/verbs normally used dynamically'.
I get your point, but people do talk about the partial rule of law, so it can be modified. Some learners take patterns of usage and even when the grammar book says things like 'we prefer', they interpret it a rule that is to be applied in every circumstance. There are descriptions and even rules about progressive forms of stative verbs, but there are exceptions.
I don't like I am owing much and will leave it for those who want to use it. Some books give it as a rule, others as a tendency, so saying its not an absolute rule seems OK to me. The other option would be to say that a single exception means there's no rule or that books are wrong, which may undermine learners' trust.