Are there/have there been examples in the UK where a major station has an adjacent annexe station with a different name whose existence is primarily to relieve the traffic in the major station?
Manchester Exchange came about for this reason.
Between 1844 and 1884, the LNWR (and its predecessor companies, the L&MR and Grand Junction) shared Victoria station with the LYR.
By the 1880s, congestion and delays to LNWR trains at Victoria meant that company had to build its own adjacent station at Exchange.
Following merger of the LNWR & LYR in 1922, the two stations were joined by the famous 11 Middle platform.
Not sure whether the LNWR would have ever considered Manchester Exchange to be an "annex" to Victoria, though.
Morecambe originally had two terminal stations, one for the Midland Railway, another for the LNWR.
Following Grouping in 1923, the stations were re-named Promenade (ex-Midland) and Euston Road (ex-LNWR) and the LMS then concentrated much of its Morecambe traffic at the better-positioned Promenade station, using Euston Road mainly for the summer overflow.
As seaside traffic by rail fell off in the late 1950s, Euston Road was closed completely during winter months, but still opened 7 days a week in summer - with a minimal service mid-week and was only busy on summer Saturdays.
Euston Road closed to passengers in September 1962. I'm not sure of the distance between Promenade & Euston Rd - it wasn't far, but probably not close enough to be strictly a cheek-by-jowl annex.