Did you obtain a first or a 2.1 in your degree? Your experience would probably make you a good candidate for firms which deal with matters such as family disputes, social work related matters, Court of Protection work and so on.
In London, firms such as Bindmans, Hodge, Jones and Allen, and others active in their types of practice might be suitable places to seek training.
A speech impediment might respond to training/therapy, and might diminish as your confidence grows. Nerves are normal, but can be managed through experience.
Not as many solicitors have qualified or are active as Solicitor Advocates as you might expect, and it's worth checking with any law firm in which you are interested what opportunities they provide for advocacy, especially in the High Court and above, and not limited to small directions hearings and the like.
Smaller firms tend to be more receptive to having Solicitor Advocates as part of the team. The economics of large law firms appear to discourage advocacy, although one or two of the very large commercial firms do get involved in advocacy. For the present, the independent Bar appears likely to continue. The big law firms have had the ability to extinguish/absorb the independent Bar since the early 1990s, but have chosen not to do so.
Starting as a solicitor and transferring to the Bar after a few years is an increasingly common route.
I notice that 4-5 Gray's Inn Square has just advertised some mini pupillages, so maybe throw your hat in that ring.
Good luck!