How are the neat sinc function shaped carriers produced in OFDM?
These sinc functions, as you've noticed, have zeros in a distance of the subcarrier spacing \$\Delta f\$.

Remember how these sincs come to be (the texts you've been reading most definitely mention that!): The sinc function is the Fourier transform of the rectangle function. Scaled to yield zeros in frequency domain every \$\Delta f\$, the width \$T\$ of that rectancgle must be \$T=\frac1{\Delta f}\$.

So, that answers your question: all your sincs are just the result of having a rectangle in time domain, and multiplying it by \$e^{j2\frac{n\cdot\Delta f}{f_\text{sample}}t}\$, so to shift it in frequency to yield the \$n\$th subcarrier. The QAM symbol is just a complex factor you multiply the result with – that  is just a constant factor and doesn't change the shape, neither in time nor frequency domain.

Now, what's \$\Delta f\$, when you think about it? In OFDM, you use the \$N\$-point DFT to divide your Nyquist bandwidth (complex!) \$f_\text{sample}\$ into \$N\$ equally large subcarriers, so \$\Delta f = \frac{f_\text{sample}}{N}\$. Therefore, the width of the rectangle \$T=\frac1{\Delta f}=N\cdot\frac{1}{f_\text{sample}} = N\cdot T_\text{sample}\$. That very simply means that the sinc shapes are just the effect of turning on a (complex) oscillation of frequency \$n\cdot\frac{f_\text{sample}}N\$ for exactly \$N\$ samples.


  Each transition from 1 symbol to the next, ...


Such a transition simply doesn't happen within one OFDM symbol: For the duration of one of these rectangles, the symbol for each subcarrier is constant. So, you use \$N\$ samples to send a single symbol, but you gain the ability to send \$N\$ symbols at once. So, nothing lost, nothing gained here!