Has this problem been solved since?
Not in the sense Feynman meant.
Approximate way to describe action of one charged part of body on another is known since Lorentz - the so-called Lorentz-Abraham-Dirac term.
What Feynman is getting at is this term works somewhat, but leads to contradictions when pushed to its consequences.
The problem of self-action of charged sphere on itself was not completely solved, because it involves modelling charged sphere in a way consistent with theory of relativity. Since in theory of relativity the sphere cannot be rigid but has internal degrees of freedom, the model gets very complicated very soon.
There are only approximate theories of charged sphere, search the works of Arthur Yaghjian and Rodrigo Medina.
Feynman and others thought there should be similar self-action term even for point particles, but this is not necessary.
Consistent theories of charged point particles were described many times long time ago, e.g. by Frenkel:
J. Frenkel, Zur Elektrodynamik punktfoermiger Elektronen, Zeits. f. Phys., 32, (1925), p. 518-534.
In English, this article also explains it concisely:
R. C. Stabler, A Possible Modification of Classical Electrodynamics, Physics Let-
ters, 8, 3, (1964), p. 185-187.
There is no self-action in this kind of theory.