you're answer to b is correct but it is T(0) not T(t)

This is not going to be very clean anyway you approach it--so have a cup of coffee and a cigarette

if s is the speed = |v|

It might be easiest to compute aN = k(ds/dt)^2

then aT = sqrt( |a|^2 - aN^2)

Where k = |T'|/|r'|

Normally you can compute the components by :

if s is the speed = |v|

Then aT = d^2s/dt^2

Without having to compute the curvature you can compute

aN = sqrt( |a|^2 - aT^2)

wherea= (8.8)^2 e^(8.8t)i+ e^tj

However for this problem d^2s/dt^2 is very cumbersome