Without actually seeing what you are doing while playing/picking up the instrument, the two things I can think of are as follows:
1) you may be unconsciously holding tension in your right arm while playing, like keeping your arm stiff and flexed while playing, which could lead to stiffness and discomfort when you put the horn down. Next time you practice, try noticing if you arm feels tense or strained, and if so try to relax through stretches and breathing to release that tension. A lot of times we carry around tension in different areas of our bodies without really noticing it because we get so used to it.
In general, you want to let your left arm support most of the weight of the instrument (or, on your lap while sitting/with the aid a lap pad if you have one) so that your right hand can be free and relaxed for all the finger dexterity we need while playing. It's a similar idea to trombone, where the goal is to have the right arm (slide arm) be relaxed and loose to freely move as needed, the weight of the horn supported by the left shoulder and arm.
2) Related to the first, but you might try some gentle arm stretches both before and after your practice sessions to help keep the muscles in your arms from getting stiff, releasing tension, and keeping flexible and relaxed. While doing stretches, remember to take slow, deep breaths and feel the stretches during the exhalation.
I want to emphasize that if you still experience any sharp pain in a specific area during a stretch, stop! You may want to consult with a physical therapist to get a more specific idea of what's going on, and specific guidance to help you out.
Willson 2900 TA-1 Euphonium - Denis Wick 4AM
Yamaha YSL-643 Trombone - Bob Reeves BrassArk 5G "Gladstone"
Yamaha YSL-8440 Trombone - Denis Wick 5BS
VMI 3301S BBb Tuba - Schilke Helleberg
Past:
York Preference 3067 Euphonium - Denis Wick 4AL
Benge 165F Trombone - Benge Marcellus
Wessex BR140 Baritone - Denis Wick 6BS
F.E. Olds Special Trombone (ca. 1941)