Problem storing Kosher Dill pickles

In general, you shouldn't trust anything with a poor seal to last outside the fridge. In the fridge, they'll last just as long as a well-sealed jar that you've just opened. Commercially processed pickles can last a year in the fridge. I wouldn't expect homemade ones to last as long but they should still be good for long enough to eat them all, at least months!

The solution in the future is simply to switch to a different canning method, for example this lower temperature one or a boiling water bath one. Here's an example recipe - you could certainly adjust the spices to match the recipe you like, but it's of course safest to stick to that trusted recipe for the bulk ingredients.

Your problem is that this canning method doesn't reliably produce good seals. The similar inversion method, mentioned by Martin in the comments (turning the jar over after filling and closing) does work better, but even that doesn't always make a good seal! See for example this USDA source, which says:

Some other methods of sealing jars call for inverting a closed, filled jar of hot product for anywhere from thirty seconds to one hour. (Inverting is turning the filled jar upside down on its lid.) While this inversion process can be successful in producing a sealed jar, it works best with very hot product. Individual variation in practicing this procedure or unexpected interruptions can result in delays between filling jars, getting lids screwed on, and inverting the jars. If the product cools down too much, the temperature of the product can become low enough to no longer be effective in sealing jars or preventing spoilage.

When the inversion process does work, the vacuum seals of filled jars still tend to be weaker than those produced by a short boiling water canning process. A larger amount of retained oxygen in the headspace may allow some mold growth if airborne molds contaminated the surface of the product as the jar was filled and closed. More complete removal of oxygen from the headspace also offers some longer protection from undesirable color and flavor changes with some types of fruit products. A weak seal may be more likely to fail during storage.

The tea filter idea works well and is exactly what I would propose.

You can buy ready-made paper or fabric tea filters that are smaller than coffee filters and more porous. If you want to use metal filters, make sure they are not reactive (like stainless steel) or they can influence the taste of the finished product.

Put your herbs in a filter bag, tie off the opening and put it into the container of your pickles. If you need a lot of herbs, you should use several tea filters, each filled with 2 teaspoons of herbs.