A tall stem with a short horizontal reach can be your friend here.
As a VERY rough check, put your elbow against the nose of the saddle and see whether your fingertips can touch the horizontal top of the drop bar. If they fall short, you will need to move the saddle forward or replace the stem with one with a shorter reach. (Or get an adjustable reach stem, if you can find one.
)
I have the opposite problem, in the sense that I am more likely to experience hand tingles with the mountain bike than with my road bikes, although putting bar extensions on the mountain bike did help a great deal.
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"Far and away the best prize that life offers is the chance to work hard at work worth doing." --Theodore Roosevelt
Capo: 1959 Modell Campagnolo, S/N 40324; 1960 Sieger (2), S/N 42624, 42597
Carlton: 1962 Franco Suisse, S/N K7911
Peugeot: 1970 UO-8, S/N 0010468
Bianchi: 1982 Campione d'Italia, S/N 1.M9914
Schwinn: 1988 Project KOM-10, S/N F804069