I think you should be safe, but depending on the abrasive pad (plastic or metal?), you might have scratched the enamel.
Try the different methods to clean a burned enamel cast iron; most methods used boiling water with baking soda.
From what I've read, the answer is that the oil will be sticky when the oil is not cooked sufficiently to finish both polymerizing and the carbon deposition. This is a chemical reaction, and like any endothermic chemical reaction takes some combination of time and heat to occur.
From Science of Cooking:
Note 3: If highly unsaturated oils are used and ...
It is a chemical quality of the oil called "iodine number". There is nothing you can do about it, it is as inherent in the oil as its smoke point. Oils with a low iodine number create hard polymers, and oils with a high iodine number create soft, sticky polymers.
If you want a hard, nonstick surface on the pan, choose the right oil. Coconut oil, Palm oil ...
I use my cast iron with about the same frequency as you, about once a month.
Rinse it with hot water to remove dust
Cook (possibly adding oil)
Scrape food off under hot water with spatula, fork, knife, Lodge scraper, etc.
Scour with stainless steel scrubber (no S.O.S. or detergent)
Heat on stove for 5m until all water evaporates
Pour kosher salt (rough) in ...
Sorry to be the bearer of bad news, but I find that when I'm searing anything in batches to avoid using cast iron as things tend to stick (unless the cast iron is "seasoned" and thus super non-stick!). I'll either use non-stick, or if I want to use the fond (for a soup or braise, etc.) I'll use an enameled cast iron dutch oven. If things get burn-y, the ...
Using slightly lower heat helps, but if you have several batches to do, mostly it's best to scrape/wipe out the pan between batches. I would not use cast iron in this case, as you point out, it is difficult to clean quickly. The fond can be scraped onto a plate and used at the end, alternately, the fond from the last batch can be used.