Propagating uncertainty with the slope of a tangent line
You have not propagated your errors properly. Note that the sum in your nominator evaluates to a small number with an error of roughly the same size (99.73±0.01 - 99.72±0.01 = 0.01±0.01). Also, error propagation cannot be done by simply adding up the errors.
Instead of calculating the error propagation manually (cumbersome, and prone to errors, hehe ;-) I think it's better to use suitable software.  Both R and Python (both are freely available, commonly used tools) can handle error propagation.
Here's how to do it with R (the code below should be reproducible, assuming you have installed R).
install.packages(&quot;errors&quot;)
library(errors)
options(errors.notation=&quot;plus-minus&quot;, errors.digits=1)
y1 &lt;- set_errors(99.726, 0.01)
y2 &lt;- set_errors(99.71753, 0.01)
t1 &lt;- set_errors(0.076, 0)
t2 &lt;- set_errors(0.061, 0)
(y1 - y2) / (t1 - t2)
# 0.6 ± 0.9

The value of the slope is 0.56 kPa/s, but since the error is 0.9 kPa/s, we should report just a single significant digit, so that makes it 0.6 ± 0.9 kPa/s.
And not rounding till the end is not wrong, in fact, it's good practice. As long as you handle your uncertainties correctly, too many significant digits (due to intermediate calculation steps or whatnot) become practically irrelevant, since your value cannot be more precise than the size of your uncertainty anyway.
Good luck!