I differenced the variable (univariate model) in question because it was not stationary. I ran the dickey-fuller unit root test to check for stationarity, and it looks like the differenced version is stationary. However, when I make the acf and pacf plots to determine the model, I get these bizarre results that I do not know how to interpret. Could someone please help me understand?
Since you provided ACF and PACF plots only for differentiated time series, so I can't tell you anything about the original one.
As you can clearly see on the plots, there's no significant autocorrelations in your time series (i.e. values for first lags are not significant). So, you have to choose $p=q=0$ in ARIMA model.
But there's something that points to possible seasonality existence - significant PACF for lags 9 and 16. However, this is only a hypothesis and it needs to be checked more precisely by using, for instance, spectral analysis. But first you need to explore ACF and PACF for more lags.
One more thing to check. What implies a non-stationarity in an original series? Maybe it was a deterministic trend that should be removed just by subtracting it instead of differentiating. Subtracting deterministic trend preserves more information in the processed time series.