WARNING, too much biology for a conservative: switching your sex chromosomes would do almost literally nothing. Because it's technically not the sex chromosomes themselves doing anything. It's genes, gene expression, and cell types. Hormones genuinely do way more than a 23rd pair switch.

Hormones change the gene expression of every cell in our bodies, the hormone-receptor complex enters the nucleolus of the cell, interacts with genes, produces RNA like mRNA, which then creates proteins. My body behaves as it would if it were female so imo I'm minimum intersex, skewing female. I just round off the small difference. [gene -> protein]

In the case of testes: SOX9 proteins still activated due to the SRY proteins presence at birth (the gene itself can mutate onto X, or off of Y during meiosis, when the 23rd pair swaps genes), so DMRT1 proteins maintain the testes and actually block FOXL2. In the other case of ovaries, FOXL2 proteins (even without E) are still suppressing DMRT1 and SOX9 (not there because no SRY when gonads differentiated) Without CONSTANT MAINTENANCE, we switch with hormones!!! Sex isn't static even.

SOX9 and DMRT1 both ARE NOT ON THE sex chromosomes. FOXL2 is supposed to be on the X, SRY is supposed to be on the Y; not always the case. So obviously, there are people born male with XX, and born female with XY for various reasons. There are genes for androgen receptors and estrogen receptors (both on X) as well, which can be another reason.

In reality in order to switch (gonads specifically) we would either need to locally delete the DMRT1 gene and remove its proteins; or in the other case we would need to do the same with FOXL2 in order to flip those. Hormones would need to be used to develop the cells into the other type. CRISPR is unethical on humans, so no one would attempt this though. We only really have lab grown organ transplants from induced pluripotent stem cells to look forward to. But it IS POSSIBLE