Relaxation oscillator period
The first equation you have models a capacitor charging from 0 to Vs. What you need to do is figure out what values of t correspond to the thresholds, and subtract those to get the time it takes to charge from one threshold to the other.

Also, you made a small mistake in your algebra.  

$$V_c = V_s\left(1-e^{\frac{-t}{RC}}\right)$$

$$\frac{V_c}{V_s} = 1-e^{\frac{-t}{RC}}$$

$$e^{\frac{-t}{RC}} = 1 - \frac{V_c}{V_s}$$

$$\frac{-t}{RC} = \ln\left(1 - \frac{V_c}{V_s}\right)$$

$$t = -R C \ln\left(1 - \frac{V_c}{V_s}\right)$$

Then you want to use

$$t_{rise} = -R C \left[ \ln\left(1 - \frac{V_{c,final}}{V_s}\right) - \ln\left(1 - \frac{V_{c,initial}}{V_s}\right) \right]$$

to get the risetime. 

For Vs = 5, initial voltage of 0.7 (from the diode) and final voltage of 3.0 (Vt+ from the datasheet), you get trise = 3.82.  I presume the actual Vt+ may be a bit less than 3 volts.  A final voltage of 2.4 volts gives a risetime of 2.5 seconds.