Without seeing your site specifically, there is no good way for us to narrow down the specific problem, however, I can tell you this:
I am making some assumptions in this answer due to lack of specifics.
Whether the description meta-tag is used within the SERP is highly dependent upon the search query and the content of the description meta-tag.
The best way to see an unfiltered result is to do a
site: search without search terms. For example:
Google will often list the home page first followed by the most important pages in order. It is highly likely you will see your description meta-tag as written. It is possible that Google will chose not to show the description meta-tag at this stage. Often this is a result of a description meta-tag that is too short or too long. The description meta-tag should be long enough to continue onto a second line in the SERPs and should not be longer than about 170 characters.
Not only do you want a long enough description meta-tag, but you want a description meta-tag that is semantically similar to the content and contains search terms that most people would use in a search query to find your page. I hate talking in terms of keywords in light of search being so semantic oriented these days, it really is not about keywords, however, for the description meta-tag this is one time you want to think in terms of keywords ie. search query terms.
The description meta-tag is one of the most important semantic clues you can give to a search engine. It must match your content. Where a search query hits your content, if the description meat-tag agrees with the semantics of the content, ie. has a search query term in the description meta-tag and hit upon in your content, Google is more likely to use your description meta-tag. If you can craft your description meta-tag well enough to cover most search queries that find your content, your description meta-tag will show more often. It is likely impossible to craft a description meta-tag that will always show, however, it can be crafted to show nearly all the time or most of the time.
Otherwise, if a description meta-tag is not crafted well, Google will likely pick up part of your content. Your one and only h1 tag can be used in place of your description meta-tag, though it is not likely that any other header tag will be. If the h1 tag is not seen as desirable, then a content snippet will be used. The good news it this. It appears that your h1 tags are good. My advice is that the h1 tag should be similar to the title tag only a bit longer and use a few of your most important semantic terms. Following that, the description meta-tag should be similar to the h1 tag only a bit long and contain as many semantic terms as makes sense without stuffing. Keep in mind that all three tags must be conversational and compelling and made for humans and not machines. These are being graded semantically.
Also keep in mind that Google and Bing do not make direct keyword matches. Term matches are a by-product of semantic analysis and sought last only to highlight the search term matches. So please do not think in terms of keywords as much as writing a good article that hits all the points. If well written, your article will be found as it should and not as you try and manage it. This is far superior and returns far more search results overall.