Hi Caroline,

That is indeed a clothes moth. Heat and cold treatments will kill the moths, but I would be wary to the damage to your fabrics those treatments can cause. 

Heat treatments require professionals to do, but cold treatments can be done yourself. Just stick infested items in the freezer for 72 hours.

To prevent reinfestation, store vulnerable materials in airtight containers. Low quality, thin plastics can be chewed through by moth larvae (e.g. plastic bags). 

I strongly recommend purchasing clothes moths traps. These are available online, and at most department stores. These traps are laced with the sex attractant used by the moths, luring them to a glue board where they are caught and killed. The chemicals present in these traps have no risk to human (or pet) health. 

Your aversion to insecticides is warranted. Spraying for clothes moth control is a waste of money and creates an unnecessary exposure to toxic chemicals. But there are other insecticides which are more efficacious against clothes moths while reducing the human health risk. Moth balls will kill some moths, but do not provide consistent control. In contrast, PDB (paradichlorobenzene) moth crystals and DDP (dichlorvos) pest strips are quite effective. They are meant to be applied in closets where children wouldn't have consistent exposure. 

But nonetheless, I understand the concern. You can manage the problem yourself with the traps, freezing items you suspect of infestation, and vigilant monitoring. 

You're probably right about freeze treatment. That is meant for infested items and is effective when an infestation is minor and in its initial stages.

The gap you described could be a clue as to why the infestation hasn't abated. Perhaps moths are sneaking into your house through that gap from a food source you are not able to access/treat. I recommend trying to seal that gap somehow. Or you could stick a small amount of moth balls, DDP, or PDB into that gap. 

I don't recommend treating the whole house. I say that because it is possible to manage this yourself without incurring the large cost from a professional treatment (along with the inconvenience/stress of having to temporarily vacate the space). To kill these moths, you have to achieve a temperature of 120F for 30 minutes. If you go this route, be wary of the damage to items in your house (even wood). I cannot recommend specific companies, nor am I aware of certain companies being better than others. But this list of pest control companies from the Better Business Bureau is a good start. 

https://www.bbb.org/us/co/denver/category/pest-control