My only experience is with Ford products. For a 20-foot enclosed trailer I suggest you'll want a bigger engine and stouter chassis; I tried towing a 24-footer with a 302-equipped E150 and it didn't cut it.
If you can find an older (mid-90s) F-250 or -350 with the diesel engine, that would work fine. Don't get the non-turbocharged engine (before 1993?), and you can probably find a good deal on the dirt-simple-to-maintain turbocharged mechanical injection pre-Powerstroke diesels ('94-95?)
Transmissions on the Fords can be an issue, mostly because they're tuned for comfort so the trans slips a lot. But if you get a good buy-in price you can afford to have it reworked. Mass Diesel near Boston can convert these things into indestructible tow beasts. If you find one with an already-good E4OD or 4R100 trans then buy the Banks Power TransCommand; it'll up the band pressures and tighten the shifts up.
If the budget allows, move up to the later 90's Powerstroke trucks. That 7.3L will tow like nothing else. They tend to retain their value, however, unless really long in miles. If you can find one within your budget, any Ford turbodiesel truck will do.
Finally, I was fairly satisfied towing with my 6.9L V10 E350. I'd suggest as late a one as you can afford, as they made several improvements in the heads over the years relating to airflow (read: more power) and spark plug bosses (earlier ones were thin, and ham-fisted spark plug replacements could result in stripped threads and plug blow-outs). The 6.9L V10 will pull a 20-footer with NO problems. My E350 had the earlier (1999) engine plus engines are further de-rated in the vans, but it pulled my 24-footer well. You had to let the speed drain a bit on long hills, and you had to be willing to let it rev, but other than when climbing the Blue Ridge Mountains (top speed 55 mph floored and revving) I never really had any issues with keeping up to speed. Had I the pickup version of the engine and/or the later higher-flow heads (post-2001?) and I would have been perfectly happy.
Best part about the V10 is that you can get a nice one CHEAP. I do mean CHEAP. Do a search on a 2002 vintage F350 crew cab long bed and I think you'll find it's within your budget.
Worst part about the V10 is fuel economy: expect 7-8 mpg while towing, up to 12-13 mpg in lightly-loaded highway cruising (not towing). But given you can get one of these for such lower money, that will more than make up for additional fuel costs. Plus, maintenance is cheaper on the gassers.
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