We investigate the nonlocal Hanle effect in high mobility two-dimensional electron systems using (Ga,Mn)As/GaAs spin Esaki diodes as spin selective contacts. Spin signals in these systems can be strongly affected by dynamic nuclear polarization, which mimics long spin-relaxation times extracted from the measured Hanle curves. Here, we introduce a method which largely suppresses these effects by using an ac injection-detection setup. This allows us to extract from the measurements realistic spin lifetimes on the order of single nanoseconds. As the detection of Hanle signals is also strongly affected by offset signals we discuss the magnetic field dependence of these background voltages observed in lateral nonlocal spin injection devices. We show how the strength of the background magnetoresistance can be minimized by choosing a proper device geometry.