"Ich soll meine Lehrerin eher fragen."
Interesting. I would have said no, at least not in U.S. English, but it would have a usage that, in this case, would mean "instead." Seems just a twinge archaic to me--but again, that may only be in U.S. English.
I'm from the UK too but would agree with helenvee. It's not used super often but "I should rather" is still valid British English.
"Sooner" versus "rather" makes a difference in meaning here, at least in US English. "Sooner" is generally a time reference (although some dialects do make it synonymous with "rather.") In this case, it would mean "I am supposed to ask my teacher [sooner than you want me to, sooner than some impending date, etc.]." Does "eher" lean one way or the other in German, or would it depend on context?
It depends on where you put the 'eher'. "Ich soll eher meine Lehrerin fragen" = "I'm supposed to rather ask my teacher (instead of the janitor whose advice wasn't that accurate)". "Ich soll meine Lehrerin eher fragen" = "I'm supposed to ask my teacher sooner" (e.g. if in the past I didn't ask any questions about a topic and therefore got totally lost. When I finally asked my question, it was already too late... so much stuff I had missed because I couldn't follow. So, in future I'm supposed to ask her sooner)
Thank you. So, does "eher" essentially modify what comes immediately after? In other words, "eher meine Lehrerin" = my teacher rather than someone else; "eher fragen" = ask sooner?
I'd say 'yes'. (I'm not 100% sure that there isn't any exception, but basically that's it)
In my not so humble opinion, neither of those is acceptable. "I am supposed to" or "I should" are ok. ;-)
It hasn't been answered here yet, so I will try my best: 'I should' is not accepted here, because 'should' infers doubt, hence it uses the subjunctive II, much like could, would, etc. Hence, "Ich sollte meine Lehrerin eher fragen" = "I should ask my teacher sooner". I think in that case 'instead' would work better than sooner, it doesn't quite sound right to me.