The characteristic feature of paraconsistent logics is their rejection of the principle of explosion, which says that from a contradiction anything follows. Dennis linked to a survey article of various logics belonging to this family (see esp. §§3.6–7). Relevant for our purposes here is a fact mentioned in §1:
Fact 1. Many paraconsistent logics validate ¬(φ ∧ ¬φ) (aka the law of non-contradiction).
This means that paraconsistent logics don't necessarily tolerate conflicting accounts of the world, i.e., there are paraconsistent logics that validate no sentence of the form φ ∧ ¬φ.
Now, there is a general philosophical view known as dialetheism, according to which there are true contradictions, i.e., some formulas of the form φ ∧ ¬φ are satisfiable! It seems to me that the 'territory' you speak of is not paraconsistent logic, but rather this more general dialetheist view. That's not to say that the two aren't related: since dialetheists tolerate contradictions, in order for their theories not to be trivialized, they must reject the principle of explosion. This makes paraconsistent logics a natural platform for those who want to create dialetheist-friendly logics.
Temporal logics, as Dennis already mentioned, are usually just modal logics in disguise. Depending on your intuitions about the structure of time, you choose as your base from a number of different modal logics. None of the modal logics I've come across have been paraconsistent, but such a combination might, of course, be possible and even turn out to have useful applications. Although the existence of non-paraconsistent temporal logics (e.g. the Priorean tense logic) already answers your question negatively, I understand your question arose from the observation that in temporal logic, a formula φ can be true at time t and be false at time t'>t and then maybe again true at time t''>t', and so on. But fortunately there is no conflict here at all. For non-dialetheist temporal logics this is generally true:
Fact 2. For any moment t and any formula φ: either t |= φ or t |= ¬φ.
This just follows from the fact that the base modal logic is classical and thus accepts the law of non-contradiction. Of course, this wouldn't generally be true for paraconsistent temporal logics and would never be true for dialetheist temporal logics.
Hope this helps. For corrections/suggestions, please leave a comment or just edit this post.