In bacteria, they produce a lot more mutations in stressful environments because it is evolutionarily advantageous. In other words, if your situation is killing you playing Russian roulette with your genes is a better bet than accepting certain doom. Maybe you produce a deadly mutation that kills you and maybe you produce a mutation that helps you survive certain doom.
My recollection is that snails can reproduce either sexually or asexually and they preferentially reproduce sexually in stressful environments and asexually in environments that make staying the same more advantageous.
Sickle Cell is protective against malaria. Sickle Cell trait is protective without causing Sickle Cell Anemia, which is a horrible condition. So one copy of the mutation and you are more likely to survive in an area where malaria is prevalent and two copies and you are jacked up, but maybe less jacked up than with malaria.
Some studies suggest that Cystic Fibrosis is a predominantly Caucasian disorder because having one copy of the gene is protective against certain disease that were sweeping through Europe at one time. Two copies tends to kill people gruesomely at young ages.
So I think generally speaking the answer is that species seem to seek mutations when what they are doing currently isn't working and seek stability when what they are doing currently is working.
Also I have read that it is believed that half or more of all human pregnancies probably end in the first two weeks and result in a heavier-than-normal period without the woman even realizing she was ever pregnant in most cases because those fetuses are simply not viable. Laying bets on "Will this novel mutation or novel combination work?" tends to get a result of "Nope. It so doesn't work, it's not worth investing precious resources in to bring the baby to term and let it be born."
We mutate more when it is "mutate or die" and less when mutating is the thing more likely to kill you.
There are a few things in here that are fun to think about.
In the situation you describe, the answer is often "yes" and that's how you end up with evolution. Let's say you have a gene that makes a protein that digests glucose. And then one day, your cell messed up when replicating and accidentally made an extra copy of that gene. Well now you have an extra copy of that gene that isn't under purifying selection. It's redundant. It can mutate but as long as you have the first copy, you're ok. And eventually it mutates away from being good at digesting glucose. It can do it a little bit, but it's not great. Maybe it's 20% as effective as it originally was. But you have another gene that's still 100% effective so you don't even notice.
Now we have a protein that really doesn't do anything bad... It just doesn't do much good either. And since it isn't subject to purifying selection, every round of replication it keeps mutating. Until all of a sudden, it mutates into something that can digest lactose. Now, you have an evolutionary advantage from a protein that first had to get bad at binding glucose, before it could benefit you. But evolution has no foresight, so it didn't know. So it took getting rid of purifying selection to make it happen. But now as you come to rely on lactose, that protein will wind up back under purifying selection and become "fixed".
So now let's consider an alternative situation. You only have one gene that can digest glucose. If it mutates to be 20% effective, you will at best grow only 20% as fast as your competitors. Maybe you even die and become an evolutionary dead end. In that case, it is an extreme disadvantage to have a protein that can't do its job reliably, and organisms that don't have a malfunctioning variant will grow better and pass on their genetic material to more offspring, until you are eventually outcompeted and go extinct.
We can also imagine another scenario. Your glucose digesting protein mutates into something that can still bind glucose, but cant digest it. Then the glucose remains stuck to the protein, producing no energy, and becomes a waste for the cell. That is actively harmful and will likely kill the cell very quickly.
Wouldn't a 20% chance of a novel, useful interaction and 80% chance of no effect be almost as good if you are producing a lot of them?