Conflicting ACF/PACF after first-difference

I have yearly data. When I do a Dickey-Fuller test it gives me insignificant results, indicating that the series are non-stationary. After differencing them the DFT tells me they are now significant and stationary.

When using an ACF/PACF plot on the original series it is obvious that an AR(1) process should be used.

When plotting ACF/PACF for the first difference I can't make that same inference again because there is no significant spike in the ACF.

There is nothing conflicting here. Differencing a unit root process removes the unit root. If the original process is AR(1) with a unit root, $$y_t=y_{t-1}+\varepsilon_t$$, the differenced process is just white noise, $$\Delta y_t=\varepsilon_t$$. This is what you see in the ACF and PACF for the differenced data.