You can cut down the search a lot by using modular arithmetic.
All primes > 3 are equal to either 1 or 5 mod 6.
98 = 2 mod 6.
The only possibilities that sum to 2 mod 6 are
1+1+1+1+5+5, 1+5+5+5+5+5, 3+1+1+1+1+1, 3+1+1+5+5+5
Consider the first option 1+1+1+1+5+5.
The only sums of 4 primes equal to 1 mod 6 that are less than 98 are
7+13+19+31 = 70
7+13+19+37 = 76
7+13+19+43 = 82
So the 2 primes equal to 2 mod 6 must sum to 16, 22, or 28.
The only options are
5+11=16
5+17=22
5+23=28
11+17=28
So we have 4 solutions, so far.
The rest is left as an exercise for the reader.
