Lead acid battery increasing current under constant voltage
Sounds normal to me.

The electrolyte in a deeply-discharged battery is very weak &mdash; almost pure water, because the sulfur is bound up as lead sulfate in the plates. This makes the internal resistance very high, resulting in a low charging current initially.

Charging the battery releases some of the sulfate into the electrolyte as sulfuric acid, gradually increasing its conductivity and the charging current.

It sounds like this battery may still be usable. If the enough of the sulfate crystallizes and becomes insoluble, then the battery is generally unrecoverable.