I would say that the high molecular weight smear will contribute to a raised background but the huge amount of amplified product will give you good readable sequencing (Sanger)
You could get good results or double sequence. As you know, agarose gels have less resolving power than polyacrylamide gels. So what might appear as a single band on an agarose gel, may contain multiple PCR products of a similar size. If your primer binds to more than one product, you'll get double sequence.
I would try sequencing a few samples first to see if you get good results. If you don't, you might want to consider reamplifying your desired product by cutting out the correct size band from the gel, eluted it, and using the product as a template for another round of PCR. Good luck.
I would advise that you gel purify out the bands i.e cut out the visible expected PCR product from the gel and purify it using a PCR purification kit, you will then elute out a single PCR product that you can send for sequencing.
The challenge with sending a PCR product with a lot of smears for sequencing is that you will have alot of background in the reaction which will lead to poor chromatograms and thus sequencing results
The uneven outline of these bands seems to be a result of the electrophoresis. It would help to see the bands of a molecular marker. The smear can be a result of overloading or high molecular fragments.