yet another burned MOSFET issue in H-bridges
Notice the fine print on DMP3056 datasheet about how RthJA of 91°C/W is measured...

Note 5: Device mounted on FR-4 substrate PC board, 2oz copper, with thermal vias to bottom layer 1inch square copper plate.

The other FET specs 131°C/W but...

(a) For a device surface mounted on 25mm x 25mm FR4 PCB with high coverage of single sided 1oz copper, in still air conditions.

In other words these specs count on having enough copper on the board to act as a heat sink, with heat flowing through the component leads.
Some special packages can have a low Rth because the chip is mounted directly on the metal lead frame which conducts heat to the board through the pins and solder. This is not the case here, this standard SOT-23 can't dissipate half a watt.
Solution: use a FET package that will handle the required power, if this is a one-off prototype, TO220 is ideal because it's much easier to solder on protoboard than SMDs.
Or get a ready-made h-bridge, these are quite cheap.
Or keep SOT23s but pick ones with much lower RdsON.
Note if the power supply current limits at 3A, you need a more powerful supply for a 3A peltier.