In some cases (no pun intended) good air coolers can outperform AIO's, your case will only fit a 120mm rad I believe at the back of the case, unless those drive cages at the front are removable where you might fit a 240.
Good air coolers are expensive though and a 120mm AIO will probably do the job for cheaper, ymmv
I don't know where you get the idea that 35C is particularly hot, it's not. If it was you wouldn't be able to use a PC in any "warmer" country without exotic cooling.
Coolers work with a delta above ambient, 20deg+ is bad, it has nothing to do with the starting ambient temperature, that's why it's a delta.
However yes a new cooler will almost definitely fix it as it sounds like the current one is faulty or just isn't up to the job.