Okay, here's a suggestion - there were lots of rocking chairs of the era, even catalogue -
so use minimum restoration - get some triple ooo steel wool, some old wash rags, old english lemon oil polish, and some mineral spirits paint thinner but NOT lacquer thinner - clean it first with the oil and rags - briskly - wipe clean, set overnite - clean it same way again, leaving a heavier coat of lemon oil, and let this oil set for a couple of hours - rub with the steel wool, wipe excess with rags - like what you see? that's all you need to do. If there are heavy dirt areas, then re-assess. GENTLY rub with the paint thinner on a rag - if that does the job, then lemon oil it again. This is my method of over 40 years experience with old furniture - I love the worn look, and this kinda' restores the sheen without taking away the look of the old wood which is why I buy these to begin with. Oh, and once you get that nice worn sheen, use some paste wax. And avoid spray wax stuff.
Congrats on a nice vintage piece. KB