The Rishonim address the first issue of why today we don't count two days each day, and they answer that since Sefiras HaOmer is only D'Rabbanan today it would be inappropriate to be machmir on counting Sefira in a way that would lead to the disrespect of Shavuos. Since we would end up counting the 49th day in doubt on the first and primary day of Shavuos, it would lead to reduce the dignity of the day. Alternatively, there are Acharonim who answer that counting by definition requires a specific and accurate reckoning, which cannot be reconciled with counting two times each day.
Regarding what was done when there was a true safek, it is impossible to know with certainty. However, according to the second reason mentioned previously the Dvar Avraham writes that they did not count at all until the messengers from Beis Din arrived to repot which day had been proclaimed Rosh Chodesh Nissan. This seems improbable because it would mean that the Amoraim in Bavel didn't count the first days of Sefiras HaOmer, and we find no mention of such a shocking fact in the Gemara. Another possibility proposed is that they in fact counted two days out of doubt (some say with and some say without a beracha), and that when they heard the accurate date they either switched to count the single accurate date or continued to count two. Alternatively, they may have only counted presuming that Rosh Chodesh was the 30th of Adar as it was most years and in our current calendar, and if neccessary adjusted their count when the heard what the Sanhedrin decided. It should be pointed out that when the Beis HaMikdash stood Sefiras HaOmer was D'Oraisa and there more reason to have counted two days.