Tumble drying them will probably have the greatest effect on making them softer. You should find that they are much softer but still just as absorbent. It's important not to over dry them, as that would leave them less soft. You want the dryer to terminate leaving just the right tiny amount of moisture in the towels.
I do use vinegar for my towels in place of regular fabric conditioner. Tests have shown that vinegar doesn't soften towels nearly as well as fabric conditioner, but nor does it reduce absorbency. Basically the more a fabric conditioner softens, the more it reduces absorbency. You have to decide which is more important to you and what balance you're prepared to accept.
You might find that a gentle, cool wash with mild liquid detergent will leave them softer, but it obviously won't clean them as well. I certainly would prefer clean towels over soft towels in this case! You might be able to still wash them at <span class="st" data-hveid="45" data-ved="0ahUKEwiYoODIsc3XAhWJKsAKHT6kASYQ4EUILTAA">60 </span><span class="st" data-hveid="45" data-ved="0ahUKEwiYoODIsc3XAhWJKsAKHT6kASYQ4EUILTAA">°C for hygiene</span> but on a gentler/shorter cycle with a higher water level and/or less agitation.
If you live in a hard water area and don't have a water softener, then it's important to add enough detergent to soften the water but without overdosing. Adding STPP (phosphate) may help -- it's readily available online.