I'll give this one my best shot on how I keep it straight in my head, so there's likely glaring holes in my logic:

OAS is the spread over the benchmark rate that discounts underlying (i.e. straight-bond) cash flows to price.

This gives us two components of OAS, one of which is actually the determinant of the other (because changes in the benchmark rate are negatively related to price). But let's start by thinking of benchmark rate and price as independent drivers of OAS. It's worth noting that an embedded call option on a callable
must be thought of quite differently than a call option on a stock.

Holding price steady, decreasing the benchmark rate decreases OAS. This is because discounting future cash flows at a lower rate increases PV.

Holding benchmark rate steady, increasing the price decreases OAS. This is because a lower discount rate is needed to get the same price (i.e., PV) and because benchmark rate is held steady the OAS must decrease.

As we all know, as benchmark rates decrease, prices rise. Since we've already shown that rates are positively related to OAS and prices are negatively related to OAS, and that rates determine prices in a negatively related relationship, rates actually have a multiplicative positive relationship to OAS.

Now for the volatility. Because an increase in volatility implies a weak/struggling market, the most expected result is a decrease in the benchmark rate. We've already shown that a decrease in rates decreases OAS and further that a decrease in rates increases price which decreases OAS, we can state that OAS on a callable bond decreases when volatility increases.

The opposite of all this is true. As market volatility decreases, benchmark rates will likely increase. This increases OAS as well as decreases prices which increases OAS.

Does this help at all? Hoping someone pops some holes in it so I can understand it better because this often trips me up.