Oregano (Origanum vulgare) is hardy in your location. Leave it planted in the ground. No need to chop it to the ground before winter. The dried plant stalks do give some winter interest as well as a visual reminder where the plant is located. Plus the plant stalks help collect snow, dired leaves, etc. providing some winter protection for the roots.

Rosemary, indoors.

FataMorgana

It's kind of off topic, but are there any suggestions for heat lamps or lights that work well?

You do not need extra heat for rosemary indoors. In fact it will overwinter better kept as cool as possible as long as it does not go too far below freezing for any length of time. As for lights I can't help as I have never used them.

I usually buy a few plants of rosemary and keep one indoors in a south-facing window year round.

The reason I do this, rather than bringing a plant indoors from my garden is to avoid bringing in any unwelcome insects. So far it has been doing great.

The plant doesn't get as big, but it provides me with a more than adequate supply for cooking.