To answer your specific question, yes, you can sell a house with owner financing you don't own outright. It is, however, easy to violate the due-on-sale clause in your current note. That's not a criminal matter, though, so not strictly "illegal". Its a civil matter and gives the lender the right to call the loan.
Your big problem is that sewer lien. I don't think any of these techniques work if the sewer lien becomes the responsibility of the new buyer.
Several ways to do it. Lease/option, land contract, wrap, or subject to. Those are in order based on the amount of control your retain.
Lease option you write two separate documents. A lease giving the tenant/buyer possession of the house. A separate option giving them the right to purchase within a specific period of time. You stay on deed until they exercise their option. If you do things right, you'll only have to evict if they don't pay.
Land contract is like the way you buy a car with a loan. You keep the deed. You and the buyer make a contract that gives them possession immediately and the property after the contract terms are fufilled. Terms are certain payments for some period of time, then one big payment. Procedures for getting them out if they don't pay. Not sure what SC does for these.
With a wrap they get a deed. You create a new mortgage. They pay you on the new mortgage, they pay the older one. If they don't pay, you have to foreclose.
With a subject to the buyer takes over payments on your mortgage. It stays in your name. If they don't pay, the lender will foreclose and take the property. That will affect your credit. Your only recourse is to sue the buyer.
All of these violate the due on sale. Yes, there are techniques to HIDE it from the lender, but they don't change the fact you have violated the clause. Will the lender call? Unlikely, but possible.
The trouble is that sewer lien. Most of these techniques rely on getting a buyer with a small down payment, maybe 3%. You're wanting that plus a $10K sewer lien. That's just not going to happen. Anyone with $13K cash to put toward a house has many options, and your house is not very attractive. You say it's not selling at $115, but would sell quickly at $100. Do you have evidence - comps, current listings - to support that number? How does your property compare? To sell quickly in a competitive situation, you have to have the sharpest house at the best price. Do you have that?
Your end up needing $106K plus $10K plus costs to avoid coming out of pocket. That sounds like an effective selling price of $118K if you avoid all the commissions and some of the closing costs. You need the buyer to cough up $13K plus costs and pay an inflated price.
If you sell to someone with a foreclosure on their record, it will take them five years to get a new mortgage.
To your question, yes, you could do this. I just don't see success given the circumstances. Either do a short sale and take the hit, or come up with a way to bring some cash to the table. I'd get this sucker priced right to get a buyer, figure out some way to add that sewer lien into the purchase price, and get your current lender to accept the short sale.