You can offer any trades to any players; and there’s no specific rules about the format that these offers take. So you can make offers in such a way that it’s basically an auction. If someone says they are willing to pay $100 for your property, then you can always ask if anyone else is willing to pay more.
So yes, you could auction your properties in this way; but really it’s just a series of offering different trades and trying to get the best deal for them. And it’s possible no one will be willing to trade with you; in which case you are stuck with the properties.
From the rules:
SELLING PROPERTY... Unimproved properties, railroads and
utilities (but not buildings) may be sold to any player as a private transaction for any amount the owner can get; however, no property
can be sold to another player if buildings are standing on any
properties of that color-group.
Note that the player who received a mortgaged property in a trade/sale must immediately either pay the mortgage price + 10% to unmortgage it, or pay just the 10% and leave it mortgaged, which means they’ll have to pay the extra 10% again if they want to unmortgage it later.
The player who mortgages property retains possession of it and no other player may secure it by lifting the mortgage from the Bank.
However, the owner may sell this mortgaged property to another player at any agreed price. If you are the new owner, you may lift the mortgage at once if you wish by paying off the mortgage plus 10% interest to the Bank. If the mortgage is not lifted at once, you must pay the Bank 10% interest when you buy the property and if you lift the mortgage later you must pay the Bank an additional 10% interest as well as the amount of the mortgage.