The F word. Uh, chord. You will not get far in the world of Dylan songs if you can't finger it. In general, what one can say about the F chord applies to all chords where you need to finger all the strings. There are four ways:
Barre chords. This requires a strong index finger, but, perhaps even more, a relaxed hand: you should not press too hard either. Your hand should know (from experience) just how hard you have to press to make all the strings sound clean, but without straining you hand.
Use your thumb. Any classical guitar teacher would kill me for saying this (and then he would kill you for following my advice), but in a sense, while they shoot me through the head, they are also shooting themselves in the foot. The reason for the "thumb always behind the neck" rule is to ensure economy of means, maximum of flexibility with a minimum of physical effort, but if you don't need to play chords like 243115, which you don't (it can be done, though...), the most economical thing is to use the thumb. The switch between C=332010 and F=133211 is much, much easier and smoother with the thumb-F than with the barre-F. Again, you don't have to push very hard to get the sound you need. (Besides, there is no way on earth you are ever going to look as cool as Keith Richards if you only play barre chords.)
Use only some strings, and/or open strings. You don't always have to finger all six strings. If you play with an emphasis on the bass, you can do with 133xxx, or if you need the full chord or a brighter sound, xx3211 is perfectly acceptable. In the latter case, you can even play x03211, since a is part of the F chord. A similar case is B flat, a terrible chord to finger the ordinary way (x13331), but much more playable as xx0331 or x5333x.
This is not limited to "standard" barre chords; a chord like A benefits strongly from a barre treatment (with or without the first string), both because it is easier, and because you can then easily switch to D/a=x04232. A half barre on the middle strings (A=x02220 with the index finger bent at the last joint) is a handy technique to have acquired.
Cheat. Try some closely related chords instead, like D minor or A minor, or allow some open strings (x03210, e.g.).
G major. This chord should be fingered with the middle, ring, and little fingers. This leaves the index finger free to do other things, or to move in position for the C chord which is very likely to follow, at least in Dylan's idiom. This is particularly true of the embellishing figure G - C/g - G (320003 - 3x2013 - 320003) which you will find all over Dylan's output. Watch Joan Baez do that with the "index-finger G" in Renaldo & Clara (or is it the Hard Rain TV special?), then go and rehearse the "pinky G" instead (I cringe everytime I watch that sequence).
And again, cheating can be a good thing. You may not need the first string: 32000x is perfectly legitimate, and should it happen to sound anyway (320000), no big harm is done - you're just playing G6 instead...
Dampening. Sometimes you have to dampen some strings. To play G11=3x3211 you need both the sixth string, which is the only g in there, and all the others, but you don't want the a on the fifth string. You have to mute it with the ring finger.
F6 is an even trickier chord. It has to be played 13x231, because you need both the c and the d. Again, the ring finger does the muting. (Another alternative is to play xx3535).
“How on earth. . . am I supposed to play 355443 from ‘In the Garden’?” Answer: you're not. You pick some of them, perhaps different strings each time. It's a bit mean of me to write a chord like that, but my intentions are good.
Incidentally (and you are never going to need this for playing Dylan), 243115 can be played with a "twisted barre", with an index finger that covers both the two 1s (second and third string) and the 2 on the sixth string. The chord can be called F#mmaj9-5. (Exercise 1: find out why. Exercise 2: find other names for it. ["Gerald" is not a legitimate answer.])