"I might sell my house if I'm obliged to" or, "I might sell my house if I have to" could sound a bit more natural.

No, it's not correct "I'm obliged to sell my house" = grammatically correct but we would not use obliged for selling a house (except possibly? in a forced legal situation) you would sell a house either voluntarily or because of your own circumstances. ill health/ unemployment/ getting married etc this should help you make (someone) legally or morally bound to do something. "doctors are obliged by law to keep patients alive while there is a chance of recovery" synonyms: require, compel, bind, make, constrain, obligate, force, put under an obligation, leave someone no option, impel, coerce, pressure, pressurize "courts are obliged to act in accordance with the strict rules of the law" do as (someone) asks or desires in order to help or please them. "oblige me by not being sorry for yourself" synonyms: do someone a favour, do someone a kindness, do someone a service, accommodate, indulge, gratify, gratify the wishes of, help, assist, serve, humour, meet the wants/needs of, put oneself out for; be kind enough to "will you oblige me by filling in this form?" be indebted or grateful. "if you can give me a few minutes of your time I'll be much obliged" synonyms: thankful, grateful, appreciative; More "If the court hearing goes in favor of the bank. I will be legally obliged to sell the house because if they repossess they will sell at auction for a low price and I may still owe them money" The only way I​ could think of to get "obliged" into a house selling sentence.

Usually "forced to" is best.. "Obligated" could work in a situation where to please someone else in the interest of fairness you feel like you should.