There are actually 4 trails in Joshua Tree National Park, the easiest of which is Pinkham Canyon. When it's in good condition, a well-driven 4x2 should be able to make it, although if you found the Cut Accross trail difficult, this will seem quite challenging. The other three require high-clearance 4x4 to complete, but large portions of each can be driven with 4x2.
There are hundreds of miles of trails with varying degrees of difficulty in and around Anza-Borrego Desert State Park (detailed
map). Trail conditions inside the park (street-legal vehicles only) are usually better than where OHV are allowed, because the OHVs tend to tear up the trails and create a lot of whoops. Trails inside the park also tend to have less traffic and less aggressive drivers, for better safety and more solitude.
There are a number of trails leading off SR78 between Scissors Crossing and Ocotillo Wells that are easily negotiated with 4x2. Off S2, Coyote Canyon is negotiable at least to second crossing and possibly to third crossing (aka Lower Willows, a major ford).
There are three trails that are currently open and that start at higher elevation, descending to the low desert: Cottonwood Canyon (Ranchita -> Yaqui Well), Rodriguez Canyon, and Oriflamme Canyon (the latter two both start from Golden Chariot Canyon, near Banner, and end in Mason Valley roughly 10 mile S of scissors crossing). These have much lower temperatures, especially in the late afternoon, but require high-clearance 4x4. Golden Chariot Canyon itself should be negotiable by 4x2. Culp Valley and Jasper trails (near Ranchita) stay at higher elevation and are negotiable by 4x2.
There 's a whole bunch of trails on the western slope of the mountains, mostly at higher elevations, many of which are suitable for 4x2. The North Main Divide trail runs along the crest of the Santa Ana Mountains (Santiago Peak). It is centrally located and suitable for 4x2. The chaparral is blooming and gorgeous. And there are still patches of snow on shaded northern aspects above 7000' in the Peninsular ranges.