Hiya, this one would be much better suited for /r/AskAnthropology to tackle.

Yes and no. This question is impossible to answer with any certainty due to a bunch of reasons I'll try to explain. On the one hand, we are talking about a time period with very few artefacts remaining and no written sources - so not much knowledge available. On the other hand, transsexuality/transgender is a very specific term and concept of one specific time and culture.

First off, however, I want to point out that English is not my first language so I apologize in advance for any strange wording.

When looking at a burial site archeologists and anthropologists have two major elements that tell them about the person or persons buried there. One is the skeleton itself, the other is the pieces of material culture that got buried together with the body - apart from the clothes the deceased was wearing those burial objects could include things like jewelry, weapons, sacrificed animals, food, toys,...

Say, we find a burial site with a male skeleton in it that shows some healed (and not so healed) injuries. The body is clad in the remnants of armor and holds a sword in its hands. So we then interpret the find as the grave of a warrior from this and that time. Or we find a female skeleton buried with many gold rings and a pouch full of needles. We can now guess that this was a rich/important woman who did a lot of sewing. We might even theorize that in this time period and culture needles were a status symbol for women because why else would they have been considered important enough to bury her with them.

Now to give you maybe the kind of answer you were expecting: Yes, there have been in Eurasia one or two findings of graves that contained a male skeleton buried with burial objects that were usually specific for the graves of women. So yes, maybe there were cavepeople born one sex and living (and getting buried) as another.

Now we get to the problems with this theory. If there is no usable DNA in a skeleton/mummy the sex of the deceased gets ascertained by the bones. E.g. the sharpness of the eye socket edge, the length of certain bones in relation to other bones, the angle of the pelvic bones. However, unlike CSI and related TV shows want to make us believe, the sex of skeletons is not always with 100% certainty assaignable. Body types are diverse and, well, intersexuality exists. So it's not always an exact science. See for example the iron age bog body Windeby 1 popularly called "Windeby Boy" aka the bog body formerly known as “Windeby Girl”… So a person whose skeleton gets classified as male might actually have been phenotypically female when alive thousands of years ago.

Another problem is that we usually only have the burial objects to tell us anything about the buried person’s life. Say the dead in the beginning with the sword and the armor was a warrior. Or maybe he was a very clumsy baker who wanted to fool the gods into thinking him a brave warrior and thus he had paid the local priest extra to send some warrior-like objects with him to eternity. And maybe the female dead with rings and the needles was actually a famous and successful assassin who used poisoned needles as her tool of trades. The at the moment rather famous “Ice Maiden” is thought to have been an important healer, storyteller or shaman due to the fact that she had very expensive and venerable burial objects with her but – unlike the male body buried near her- no weapons of any kind. BTW I once read an archeological study about said male body where it was theorized that his name had been the Pazyrik word for fox (or maybe some Latinized version of it because who needs reason) simply due to the fact that one of his burial objects was a stripe of fox fur. So burial objects can be only circumstantial evidence and very open to interpretation.

Now the other part of the problem: the concept of transsexuality is a very specific modern and us-european one. It is already very difficult to apply to other cultures of the same time. How do you translate the western concept of transsexuality or homosexuality into a cultural mindset that thinks in three sexes and four genders? And it is very important to keep in mind that social constructs like gender are based in their specific cultures and even within those ware never static but constantly in flux. Think earrings being considered at one time extremely manly and at another a sure sign of girlishness and homosexuality. While we as a society right now consider the two-sexes-model the ultimate biological truth only some centuries ago we actually had a one-sex-model in Europe – everyone was considered biologically male but either well-done or failed and thus forced to give birth. We have sources from medieval times warning girls not spread their legs too far apart when playing because otherwise their male member might drop and turn from an innie to an outie, making them fully done boys. The quote goes that the past is a different land and in archeology/anthropology it is very important to not impose one's own cultural norms and rules on the context of one's findings.

For a very long time our idea of the 'stone age' was that men were hunters, ruled the clans, fought in battles, painted the cave walls and did the rituals while women were, well, pregnant constantly, doing the cooking and maybe gathering berries sometimes. But this concept of how life and society must have been is 90% extrapolation biased by our own culture and cultural norms and only maybe 10% actual artifacts. Turns out that oops.

To add on more level of confusion: it is a common practice in some shamanic cultures for practitioners to ritualistically switch their gender to signify the separation/differentiation of their everyday persona from their religious ‘messenger between the worlds’-role in society. So instead of being gender role signifiers the aforementioned untypically burial objects might actually have been signifier of a religious role.

Tl;dr: we don’t have the contextual knowledge to be sure one way or the other, are often culturally biased and social constructs are not applicable 1:1 to different societies/times.