If there was no way of proving that dinosaurs existed would it still be true that they did?

Truth is truth regardless of who knows it, or whether nobody seems to know it.

Can there be truths no evidence of which exists?

In the sense of transient facts that are of no importance, maybe (such as instantaneous the position of every particle in the universe down to femto-meter precision, progressed into the future, with certain decisions made such that the earlier state is indistinguishable from at least one imperceptibly different alternative, it could be said that such a state is no longer evident)--but be careful not to conflate such facts with things for which merely no properly attributed evidence has yet been witnessed by a party. Lack of time or adequate measurement tools and data could be one reason for the inability to trace evidence of an extant truth. So this question is really about ignorance. It is the same as "if a tree falls in the forest and nobody hears it, did it still fall?" The whole universe provides an infinite search space, and so lacking outright omniscience on the subject, an agent cannot distinguish between something for which evidence does not exist and something for which evidence exists but he has not yet found or associated it.

A key point here: A person who is more perceptive or diligent may have found evidence for something that is dismissed by others due to their impatience or unbelief. This is an exceedingly common phenomenon, and it occurs every time someone resists learning by pretending he already knows something is untrue or does not exist when in fact he has not performed the experimentation necessary to know for himself (or else he is being dishonest about what he does know).

It is equivalent to the following fallacy:

Proving Peter: "If you do ABC, XYZ will happen."

Dubious Duke: "Oh yeah? Well I didn't do ABC, and XYZ didn't happen, so you're wrong. XYZ never happens."

Duke is wrong. He has not done ABC, meaning he has not paid the price to know that XYZ is true. He is not in a position to dispute Peter's testimony. If the promise is that doing ABC eventually leads to XYZ, then the claim is not falsifiable (it can never be proven false), however it is "true-ifiable" or verifiable, because any person who has encountered the result can confirm that the original claim it is true, and is unable to contest the claim based on personal experience and the outcome of the experiment.

I know the answer is "no" in general due to Gödel's Theory of Incompleteness

Do you mean "yes" in the sense that an unprovable but true statement exists? Gödel's Theorem applies to certain man-made formalisms of logic, not necessarily to reality or the universe at large. It essentially states that no system of logic built on a certain set of axioms can be simultaneously consistent (no answer to a particular question is both yes and no) and complete (all questions that can be asked in the logic system can also be answered in the same logic system). However, this finding is indistinguishable from the possibility that one of the axioms included in the definition of such formal logic systems is itself inconsistent, and is therefore not a robust statement on the possibility of proving things in general in the real world.

I mean this question in a more real-world sense (i.e. scientific sense).

In other words, I am talking about empirical rather than mathematical truths

Truth is truth. The label we apply to it does nothing to change its truth value. As highlighted above, an honest view of the subject admits the possibility that one of the axioms commonly accepted in such logical systems is responsible for introducing inconsistency. By the definition of reality, any inconsistency with reality is the failure of our models, and not of reality.