Euclid Algorithm to Find Muliplicative Inverse

What you have found indeed is that $-3\equiv 26$ is the multiplicative inverse of $19$ $\mod 29$.

Reducing your backtracking result modulo $29$, it becomes $$ 1\equiv -3\cdot 19\pmod{29} $$ Which is to say, the multiplicative inverse of $19$ is $-3$.

You're almost there! Multiply both sides by $-3$ and you have $$-57x\equiv -3\pmod {29}\\x\equiv-3\equiv26\pmod{29}$$