The direct answer to your question is “maybe”. They aren’t quite sure, which is interesting considering these are words that are relatively new to English. If you look them up in the OED, they present some theories couched in terms of may and might, and present conflicting evidence.
The OED says of the verb sway that it is properly two different words from the standpoint of its etymology:
- ME. sweȝe (14th c.), conjugated strong and weak, also swye, to go, move (cf. ME. forsueie to go astray), may have been a native word orig. of the OE. type *sweʒan, (3 pres. ind. *swiʒeþ), pa. t. *swæʒ, parallel to OE. weʒan to move, carry, weigh, (wiʒeþ), wæʒ, ME. weȝe, occas. wye, pa. t. weȝe, wei(ȝ), wei(e)de. (Cf. also the parallelism of swag and wag, sweight and weight.)
Formally, sweʒe might also be ad. ONor. sveigja to bend (a bow), swing (a distaff), etc., give way, yield (cf. sveigr switch, twig), causative vb. f. svig-, in svig bend, curve, svigi switch, svigna to give way; but the ME. and ONor. verbs do not agree in sense.
- The modern sway dates only from c 1500, and agrees in form and sense with, and appears to be ad., LG. swâjen to be moved hither and thither by the wind (whence Sw. svaja to swing, Da. svaie to move to and fro, G. schwaien, schweien), Dutch zwaaien to swing, wave, walk totteringly, slant, bevel.
The verb swag is now chiefly dialectic. The OED says of its etymology:
The existence of this verb is perh. attested for the 15th cent. in swaggyng (s.v. swagging vbl. sb. note), and in swage v.2 Its immediate source is uncertain, but it is prob. Scandinavian: cf. Norw. dial. svagga and svaga to sway (see sway v. etym.).
The English word might correspond to a Scandinavian form of either type (with -gg- or -g-), according to dialect; cf., on the one hand, nag v. (Norw., Sw. nagga), sag v. (Norw. dial. sagga), wag v. (MSw. wagga); on the other, drag v. (ONor. draga), flag sb.2 (Icel. flag, ONor. flaga), snag (Norw. dial. snag, snage); also Sc. swaw = undulating or swinging motion, and flaw sb.1 (ONor. flaga).
The verb swagger is apparently derived from the verb swag above, and is also quite new, first appearing in Shakespeare. The OED provides a note from a 1598 publication that remarks that:
Swaggering is a new worde amongst them, and rounde headed custome giues it priuiledge with much imitation, being created as it were by a naturall Prosopopeia without etimologie or deriuation.