Yes, you can use them with a magic weapon, so long as it is a melee weapon. And you can use them in the same turn. Searing smite is cast as a bonus action prior to the attack, and divine smite is used as part of the attack.

Starting at 2nd level, when you hit a creature with a melee weapon attack, you can expend one spell slot to deal radiant damage to the target, in addition to the weapon's damage. The extra damage is 2d8 for a 1st-level spell slot, plus 1d8 for each spell level higher than 1st, to a maximum of 5d8. The damage increases by 1d8 if the target is an undead or a fiend.

The next time you hit a creature with a melee weapon attack during the spell's duration, your weapon flares with white-hot intensity, and the attack deals an extra 1d6 fire damage to the target and causes the target to ignite in flames. At the start of each of its turns until the spell ends, the target must make a Constitution saving throw. On a failed save, it takes 1d6 fire damage. On a successful save, the spell ends. If the target or a creature within 5 feet of it uses an action to put out the flames, or if some other effect douses the flames (such as the target being submerged in water), the spell ends.